Precalculus Examples

${(x - 5)}^{2} + {(y - 1)}^{2} > 16$ , ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1

Simplify the first inequality.

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Step 1.1

Subtract ${(y - 1)}^{2}$ from both sides of the inequality.

${(x - 5)}^{2} > 16 - {(y - 1)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.2

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{{(x - 5)}^{2}} > \sqrt{16 - {(y - 1)}^{2}}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.3

Simplify the equation.

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Step 1.3.1

Simplify the left side.

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Step 1.3.1.1

Pull terms out from under the radical.

$| x - 5 | > \sqrt{16 - {(y - 1)}^{2}}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.3.2

Simplify the right side.

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Step 1.3.2.1

Simplify $\sqrt{16 - {(y - 1)}^{2}}$ .

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Step 1.3.2.1.1

Rewrite $16$ as $4^{2}$ .

$| x - 5 | > \sqrt{4^{2} - {(y - 1)}^{2}}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.3.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = y - 1$ .

$| x - 5 | > \sqrt{(4 + y - 1) (4 - (y - 1))}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.3.2.1.3

Simplify.

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Step 1.3.2.1.3.1

Subtract $1$ from $4$ .

$| x - 5 | > \sqrt{(y + 3) (4 - (y - 1))}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.3.2.1.3.2

Apply the distributive property.

$| x - 5 | > \sqrt{(y + 3) (4 - y + 1)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.3.2.1.3.3

Multiply $- 1$ by $- 1$ .

$| x - 5 | > \sqrt{(y + 3) (4 - y + 1)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.3.2.1.3.4

Add $4$ and $1$ .

$| x - 5 | > \sqrt{(y + 3) (- y + 5)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.4

Write $| x - 5 | > \sqrt{(y + 3) (- y + 5)}$ as a piecewise.

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Step 1.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x - 5 \geq 0$

Step 1.4.2

Add $5$ to both sides of the inequality.

$x \geq 5$

Step 1.4.3

In the piece where $x - 5$ is non-negative, remove the absolute value.

$x - 5 > \sqrt{(y + 3) (- y + 5)}$

Step 1.4.4

Find the domain of $x - 5 > \sqrt{(y + 3) (- y + 5)}$ and find the intersection with $x \geq 5$ .

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Step 1.4.4.1

Find the domain of $x - 5 > \sqrt{(y + 3) (- y + 5)}$ .

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Step 1.4.4.1.1

Set the radicand in $\sqrt{(y + 3) (- y + 5)}$ greater than or equal to $0$ to find where the expression is defined.

$(y + 3) (- y + 5) \geq 0$

Step 1.4.4.1.2

Solve for $x$ .

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Step 1.4.4.1.2.1

Simplify $(y + 3) (- y + 5)$ .

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Step 1.4.4.1.2.1.1

Expand $(y + 3) (- y + 5)$ using the FOIL Method.

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Step 1.4.4.1.2.1.1.1

Apply the distributive property.

$y (- y + 5) + 3 (- y + 5) \geq 0$

Step 1.4.4.1.2.1.1.2

Apply the distributive property.

$y (- y) + y \cdot 5 + 3 (- y + 5) \geq 0$

Step 1.4.4.1.2.1.1.3

Apply the distributive property.

$y (- y) + y \cdot 5 + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.4.1.2.1.2

Simplify and combine like terms.

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Step 1.4.4.1.2.1.2.1

Simplify each term.

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Step 1.4.4.1.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$- y \cdot y + y \cdot 5 + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.4.1.2.1.2.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 1.4.4.1.2.1.2.1.2.1

Move $y$ .

$- (y \cdot y) + y \cdot 5 + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.4.1.2.1.2.1.2.2

Multiply $y$ by $y$ .

$- y^{2} + y \cdot 5 + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.4.1.2.1.2.1.3

Move $5$ to the left of $y$ .

$- y^{2} + 5 \cdot y + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.4.1.2.1.2.1.4

Multiply $- 1$ by $3$ .

$- y^{2} + 5 y - 3 y + 3 \cdot 5 \geq 0$

Step 1.4.4.1.2.1.2.1.5

Multiply $3$ by $5$ .

$- y^{2} + 5 y - 3 y + 15 \geq 0$

Step 1.4.4.1.2.1.2.2

Subtract $3 y$ from $5 y$ .

$- y^{2} + 2 y + 15 \geq 0$

Step 1.4.4.1.2.2

Convert the inequality to an equation.

$- y^{2} + 2 y + 15 = 0$

Step 1.4.4.1.2.3

Factor the left side of the equation.

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Step 1.4.4.1.2.3.1

Factor $- 1$ out of $- y^{2} + 2 y + 15$ .

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Step 1.4.4.1.2.3.1.1

Factor $- 1$ out of $- y^{2}$ .

$- (y^{2}) + 2 y + 15 = 0$

Step 1.4.4.1.2.3.1.2

Factor $- 1$ out of $2 y$ .

$- (y^{2}) - (- 2 y) + 15 = 0$

Step 1.4.4.1.2.3.1.3

Rewrite $15$ as $- 1 (- 15)$ .

$- (y^{2}) - (- 2 y) - 1 \cdot - 15 = 0$

Step 1.4.4.1.2.3.1.4

Factor $- 1$ out of $- (y^{2}) - (- 2 y)$ .

$- (y^{2} - 2 y) - 1 \cdot - 15 = 0$

Step 1.4.4.1.2.3.1.5

Factor $- 1$ out of $- (y^{2} - 2 y) - 1 (- 15)$ .

$- (y^{2} - 2 y - 15) = 0$

Step 1.4.4.1.2.3.2

Factor.

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Step 1.4.4.1.2.3.2.1

Factor $y^{2} - 2 y - 15$ using the AC method.

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Step 1.4.4.1.2.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 1.4.4.1.2.3.2.1.2

Write the factored form using these integers.

$- ((y - 5) (y + 3)) = 0$

Step 1.4.4.1.2.3.2.2

Remove unnecessary parentheses.

$- (y - 5) (y + 3) = 0$

Step 1.4.4.1.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 5 = 0$

$y + 3 = 0$

Step 1.4.4.1.2.5

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 1.4.4.1.2.5.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Step 1.4.4.1.2.5.2

Add $5$ to both sides of the equation.

$y = 5$

Step 1.4.4.1.2.6

Set $y + 3$ equal to $0$ and solve for $y$ .

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Step 1.4.4.1.2.6.1

Set $y + 3$ equal to $0$ .

$y + 3 = 0$

Step 1.4.4.1.2.6.2

Subtract $3$ from both sides of the equation.

$y = - 3$

Step 1.4.4.1.2.7

The final solution is all the values that make $- (y - 5) (y + 3) = 0$ true.

$y = 5, - 3$

Step 1.4.4.1.2.8

Use each root to create test intervals.

$y < - 3$

$- 3 < y < 5$

$y > 5$

Step 1.4.4.1.2.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 1.4.4.1.2.9.1

Test a value on the interval $y < - 3$ to see if it makes the inequality true.

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Step 1.4.4.1.2.9.1.1

Choose a value on the interval $y < - 3$ and see if this value makes the original inequality true.

$y = - 6$

Step 1.4.4.1.2.9.1.2

Replace $y$ with $- 6$ in the original inequality.

$((- 6) + 3) (- (- 6) + 5) \geq 0$

Step 1.4.4.1.2.9.1.3

The left side $- 33$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.4.4.1.2.9.2

Test a value on the interval $- 3 < y < 5$ to see if it makes the inequality true.

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Step 1.4.4.1.2.9.2.1

Choose a value on the interval $- 3 < y < 5$ and see if this value makes the original inequality true.

$y = 0$

Step 1.4.4.1.2.9.2.2

Replace $y$ with $0$ in the original inequality.

$((0) + 3) (- (0) + 5) \geq 0$

Step 1.4.4.1.2.9.2.3

The left side $15$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.4.4.1.2.9.3

Test a value on the interval $y > 5$ to see if it makes the inequality true.

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Step 1.4.4.1.2.9.3.1

Choose a value on the interval $y > 5$ and see if this value makes the original inequality true.

$y = 8$

Step 1.4.4.1.2.9.3.2

Replace $y$ with $8$ in the original inequality.

$((8) + 3) (- (8) + 5) \geq 0$

Step 1.4.4.1.2.9.3.3

The left side $- 33$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.4.4.1.2.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$y < - 3$ False

$- 3 < y < 5$ True

$y > 5$ False

$y < - 3$ False

$- 3 < y < 5$ True

$y > 5$ False

Step 1.4.4.1.2.10

The solution consists of all of the true intervals.

$- 3 \leq y \leq 5$

Step 1.4.4.1.3

The domain is all values of $x$ that make the expression defined.

$[- 3, 5]$

Step 1.4.4.2

Find the intersection of $x \geq 5$ and $[- 3, 5]$ .

$x = 5$

Step 1.4.5

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x - 5 < 0$

Step 1.4.6

Add $5$ to both sides of the inequality.

$x < 5$

Step 1.4.7

In the piece where $x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 5) > \sqrt{(y + 3) (- y + 5)}$

Step 1.4.8

Find the domain of $- (x - 5) > \sqrt{(y + 3) (- y + 5)}$ and find the intersection with $x < 5$ .

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Step 1.4.8.1

Find the domain of $- (x - 5) > \sqrt{(y + 3) (- y + 5)}$ .

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Step 1.4.8.1.1

Set the radicand in $\sqrt{(y + 3) (- y + 5)}$ greater than or equal to $0$ to find where the expression is defined.

$(y + 3) (- y + 5) \geq 0$

Step 1.4.8.1.2

Solve for $x$ .

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Step 1.4.8.1.2.1

Simplify $(y + 3) (- y + 5)$ .

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Step 1.4.8.1.2.1.1

Expand $(y + 3) (- y + 5)$ using the FOIL Method.

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Step 1.4.8.1.2.1.1.1

Apply the distributive property.

$y (- y + 5) + 3 (- y + 5) \geq 0$

Step 1.4.8.1.2.1.1.2

Apply the distributive property.

$y (- y) + y \cdot 5 + 3 (- y + 5) \geq 0$

Step 1.4.8.1.2.1.1.3

Apply the distributive property.

$y (- y) + y \cdot 5 + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.8.1.2.1.2

Simplify and combine like terms.

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Step 1.4.8.1.2.1.2.1

Simplify each term.

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Step 1.4.8.1.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$- y \cdot y + y \cdot 5 + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.8.1.2.1.2.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 1.4.8.1.2.1.2.1.2.1

Move $y$ .

$- (y \cdot y) + y \cdot 5 + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.8.1.2.1.2.1.2.2

Multiply $y$ by $y$ .

$- y^{2} + y \cdot 5 + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.8.1.2.1.2.1.3

Move $5$ to the left of $y$ .

$- y^{2} + 5 \cdot y + 3 (- y) + 3 \cdot 5 \geq 0$

Step 1.4.8.1.2.1.2.1.4

Multiply $- 1$ by $3$ .

$- y^{2} + 5 y - 3 y + 3 \cdot 5 \geq 0$

Step 1.4.8.1.2.1.2.1.5

Multiply $3$ by $5$ .

$- y^{2} + 5 y - 3 y + 15 \geq 0$

Step 1.4.8.1.2.1.2.2

Subtract $3 y$ from $5 y$ .

$- y^{2} + 2 y + 15 \geq 0$

Step 1.4.8.1.2.2

Convert the inequality to an equation.

$- y^{2} + 2 y + 15 = 0$

Step 1.4.8.1.2.3

Factor the left side of the equation.

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Step 1.4.8.1.2.3.1

Factor $- 1$ out of $- y^{2} + 2 y + 15$ .

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Step 1.4.8.1.2.3.1.1

Factor $- 1$ out of $- y^{2}$ .

$- (y^{2}) + 2 y + 15 = 0$

Step 1.4.8.1.2.3.1.2

Factor $- 1$ out of $2 y$ .

$- (y^{2}) - (- 2 y) + 15 = 0$

Step 1.4.8.1.2.3.1.3

Rewrite $15$ as $- 1 (- 15)$ .

$- (y^{2}) - (- 2 y) - 1 \cdot - 15 = 0$

Step 1.4.8.1.2.3.1.4

Factor $- 1$ out of $- (y^{2}) - (- 2 y)$ .

$- (y^{2} - 2 y) - 1 \cdot - 15 = 0$

Step 1.4.8.1.2.3.1.5

Factor $- 1$ out of $- (y^{2} - 2 y) - 1 (- 15)$ .

$- (y^{2} - 2 y - 15) = 0$

Step 1.4.8.1.2.3.2

Factor.

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Step 1.4.8.1.2.3.2.1

Factor $y^{2} - 2 y - 15$ using the AC method.

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Step 1.4.8.1.2.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 1.4.8.1.2.3.2.1.2

Write the factored form using these integers.

$- ((y - 5) (y + 3)) = 0$

Step 1.4.8.1.2.3.2.2

Remove unnecessary parentheses.

$- (y - 5) (y + 3) = 0$

Step 1.4.8.1.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 5 = 0$

$y + 3 = 0$

Step 1.4.8.1.2.5

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 1.4.8.1.2.5.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Step 1.4.8.1.2.5.2

Add $5$ to both sides of the equation.

$y = 5$

Step 1.4.8.1.2.6

Set $y + 3$ equal to $0$ and solve for $y$ .

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Step 1.4.8.1.2.6.1

Set $y + 3$ equal to $0$ .

$y + 3 = 0$

Step 1.4.8.1.2.6.2

Subtract $3$ from both sides of the equation.

$y = - 3$

Step 1.4.8.1.2.7

The final solution is all the values that make $- (y - 5) (y + 3) = 0$ true.

$y = 5, - 3$

Step 1.4.8.1.2.8

Use each root to create test intervals.

$y < - 3$

$- 3 < y < 5$

$y > 5$

Step 1.4.8.1.2.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 1.4.8.1.2.9.1

Test a value on the interval $y < - 3$ to see if it makes the inequality true.

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Step 1.4.8.1.2.9.1.1

Choose a value on the interval $y < - 3$ and see if this value makes the original inequality true.

$y = - 6$

Step 1.4.8.1.2.9.1.2

Replace $y$ with $- 6$ in the original inequality.

$((- 6) + 3) (- (- 6) + 5) \geq 0$

Step 1.4.8.1.2.9.1.3

The left side $- 33$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.4.8.1.2.9.2

Test a value on the interval $- 3 < y < 5$ to see if it makes the inequality true.

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Step 1.4.8.1.2.9.2.1

Choose a value on the interval $- 3 < y < 5$ and see if this value makes the original inequality true.

$y = 0$

Step 1.4.8.1.2.9.2.2

Replace $y$ with $0$ in the original inequality.

$((0) + 3) (- (0) + 5) \geq 0$

Step 1.4.8.1.2.9.2.3

The left side $15$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 1.4.8.1.2.9.3

Test a value on the interval $y > 5$ to see if it makes the inequality true.

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Step 1.4.8.1.2.9.3.1

Choose a value on the interval $y > 5$ and see if this value makes the original inequality true.

$y = 8$

Step 1.4.8.1.2.9.3.2

Replace $y$ with $8$ in the original inequality.

$((8) + 3) (- (8) + 5) \geq 0$

Step 1.4.8.1.2.9.3.3

The left side $- 33$ is less than the right side $0$ , which means that the given statement is false.

False

Step 1.4.8.1.2.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$y < - 3$ False

$- 3 < y < 5$ True

$y > 5$ False

$y < - 3$ False

$- 3 < y < 5$ True

$y > 5$ False

Step 1.4.8.1.2.10

The solution consists of all of the true intervals.

$- 3 \leq y \leq 5$

Step 1.4.8.1.3

The domain is all values of $x$ that make the expression defined.

$[- 3, 5]$

Step 1.4.8.2

Find the intersection of $x < 5$ and $[- 3, 5]$ .

$- 3 \leq x < 5$

Step 1.4.9

Write as a piecewise.

${\begin{cases} x - 5 > \sqrt{(y + 3) (- y + 5)} & x = 5 \\ - (x - 5) > \sqrt{(y + 3) (- y + 5)} & - 3 \leq x < 5 \end{cases}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.4.10

Simplify $- (x - 5) > \sqrt{(y + 3) (- y + 5)}$ .

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Step 1.4.10.1

Apply the distributive property.

${\begin{cases} x - 5 > \sqrt{(y + 3) (- y + 5)} & x = 5 \\ - x + 5 > \sqrt{(y + 3) (- y + 5)} & - 3 \leq x < 5 \end{cases}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.4.10.2

Multiply $- 1$ by $- 5$ .

${\begin{cases} x - 5 > \sqrt{(y + 3) (- y + 5)} & x = 5 \\ - x + 5 > \sqrt{(y + 3) (- y + 5)} & - 3 \leq x < 5 \end{cases}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5

Solve $x - 5 > \sqrt{(y + 3) (- y + 5)}$ when $x = 5$ .

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Step 1.5.1

Solve $x - 5 > \sqrt{(y + 3) (- y + 5)}$ for $y$ .

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Step 1.5.1.1

Rewrite so $y$ is on the left side of the inequality.

$\sqrt{(y + 3) (- y + 5)} < x - 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{(y + 3) (- y + 5)}}^{2} < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3

Simplify each side of the inequality.

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Step 1.5.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(y + 3) (- y + 5)}$ as ${((y + 3) (- y + 5))}^{\frac{1}{2}}$ .

${({((y + 3) (- y + 5))}^{\frac{1}{2}})}^{2} < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2

Simplify the left side.

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Step 1.5.1.3.2.1

Simplify ${({((y + 3) (- y + 5))}^{\frac{1}{2}})}^{2}$ .

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Step 1.5.1.3.2.1.1

Multiply the exponents in ${({((y + 3) (- y + 5))}^{\frac{1}{2}})}^{2}$ .

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Step 1.5.1.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${((y + 3) (- y + 5))}^{\frac{1}{2} \cdot 2} < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.5.1.3.2.1.1.2.1

Cancel the common factor.

${((y + 3) (- y + 5))}^{\frac{1}{2} \cdot 2} < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.1.2.2

Rewrite the expression.

$(y + 3) (- y + 5) < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.2

Expand $(y + 3) (- y + 5)$ using the FOIL Method.

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Step 1.5.1.3.2.1.2.1

Apply the distributive property.

$y (- y + 5) + 3 (- y + 5) < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.2.2

Apply the distributive property.

$y (- y) + y \cdot 5 + 3 (- y + 5) < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.2.3

Apply the distributive property.

$y (- y) + y \cdot 5 + 3 (- y) + 3 \cdot 5 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.3

Simplify and combine like terms.

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Step 1.5.1.3.2.1.3.1

Simplify each term.

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Step 1.5.1.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$- y \cdot y + y \cdot 5 + 3 (- y) + 3 \cdot 5 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 1.5.1.3.2.1.3.1.2.1

Move $y$ .

$- (y \cdot y) + y \cdot 5 + 3 (- y) + 3 \cdot 5 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.3.1.2.2

Multiply $y$ by $y$ .

$- y^{2} + y \cdot 5 + 3 (- y) + 3 \cdot 5 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.3.1.3

Move $5$ to the left of $y$ .

$- y^{2} + 5 \cdot y + 3 (- y) + 3 \cdot 5 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.3.1.4

Multiply $- 1$ by $3$ .

$- y^{2} + 5 y - 3 y + 3 \cdot 5 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.3.1.5

Multiply $3$ by $5$ .

$- y^{2} + 5 y - 3 y + 15 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.3.2

Subtract $3 y$ from $5 y$ .

$- y^{2} + 2 y + 15 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.2.1.4

Simplify.

$- y^{2} + 2 y + 15 < {(x - 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.3

Simplify the right side.

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Step 1.5.1.3.3.1

Simplify ${(x - 5)}^{2}$ .

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Step 1.5.1.3.3.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$- y^{2} + 2 y + 15 < (x - 5) (x - 5)$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.3.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.5.1.3.3.1.2.1

Apply the distributive property.

$- y^{2} + 2 y + 15 < x (x - 5) - 5 (x - 5)$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.3.1.2.2

Apply the distributive property.

$- y^{2} + 2 y + 15 < x \cdot x + x \cdot - 5 - 5 (x - 5)$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.3.1.2.3

Apply the distributive property.

$- y^{2} + 2 y + 15 < x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.3.1.3

Simplify and combine like terms.

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Step 1.5.1.3.3.1.3.1

Simplify each term.

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Step 1.5.1.3.3.1.3.1.1

Multiply $x$ by $x$ .

$- y^{2} + 2 y + 15 < x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.3.1.3.1.2

Move $- 5$ to the left of $x$ .

$- y^{2} + 2 y + 15 < x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.3.1.3.1.3

Multiply $- 5$ by $- 5$ .

$- y^{2} + 2 y + 15 < x^{2} - 5 x - 5 x + 25$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.3.3.1.3.2

Subtract $5 x$ from $- 5 x$ .

$- y^{2} + 2 y + 15 < x^{2} - 10 x + 25$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4

Solve for $y$ .

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Step 1.5.1.4.1

Move all terms to the left side of the equation and simplify.

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Step 1.5.1.4.1.1

Move all the expressions to the left side of the equation.

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Step 1.5.1.4.1.1.1

Subtract $x^{2}$ from both sides of the inequality.

$- y^{2} + 2 y + 15 - x^{2} < - 10 x + 25$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.1.1.2

Add $10 x$ to both sides of the inequality.

$- y^{2} + 2 y + 15 - x^{2} + 10 x < 25$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.1.1.3

Subtract $25$ from both sides of the inequality.

$- y^{2} + 2 y + 15 - x^{2} + 10 x - 25 < 0$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.1.2

Subtract $25$ from $15$ .

$- y^{2} + 2 y - x^{2} + 10 x - 10 < 0$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.2

Convert the inequality to an equation.

$- y^{2} + 2 y - x^{2} + 10 x - 10 = 0$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.4

Substitute the values $a = - 1$ , $b = 2$ , and $c = - x^{2} + 10 x - 10$ into the quadratic formula and solve for $y$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 1 \cdot (- x^{2} + 10 x - 10))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5

Simplify.

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Step 1.5.1.4.5.1

Simplify the numerator.

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Step 1.5.1.4.5.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.4

Simplify.

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Step 1.5.1.4.5.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.4.2

Multiply $10$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.4.3

Multiply $4$ by $- 10$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x - 40}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.5

Subtract $40$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6

Rewrite $- 4 x^{2} + 40 x - 36$ in a factored form.

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Step 1.5.1.4.5.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x - 36$ .

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Step 1.5.1.4.5.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.1.2

Factor $4$ out of $40 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.1.3

Factor $4$ out of $- 36$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (- 9)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.2

Factor by grouping.

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Step 1.5.1.4.5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 10$ .

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Step 1.5.1.4.5.1.6.2.1.1

Factor $10$ out of $10 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.2.1.2

Rewrite $10$ as $1$ plus $9$

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (1 + 9) x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 1 x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.5.1.4.5.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 2 \pm \sqrt{4 ((- x^{2} + x) + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 2 \pm \sqrt{4 (x (- x + 1) - 9 (- x + 1))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{- 2 \pm \sqrt{4 ((- x + 1) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

$y = \frac{- 2 \pm \sqrt{4 (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.7

Rewrite $4 (- x + 1) (x - 9)$ as $2^{2} ((- x + 1^{2}) (x - 9))$ .

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Step 1.5.1.4.5.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.7.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.7.3

Add parentheses.

$y = \frac{- 2 \pm \sqrt{2^{2} ((- x + 1^{2}) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.1.9

One to any power is one.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.2

Multiply $2$ by $- 1$ .

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ .

$y = 1 \pm \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.5.1.4.6.1

Simplify the numerator.

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Step 1.5.1.4.6.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.4

Simplify.

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Step 1.5.1.4.6.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.4.2

Multiply $10$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.4.3

Multiply $4$ by $- 10$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x - 40}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.5

Subtract $40$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6

Rewrite $- 4 x^{2} + 40 x - 36$ in a factored form.

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Step 1.5.1.4.6.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x - 36$ .

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Step 1.5.1.4.6.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.1.2

Factor $4$ out of $40 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.1.3

Factor $4$ out of $- 36$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (- 9)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.2

Factor by grouping.

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Step 1.5.1.4.6.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 10$ .

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Step 1.5.1.4.6.1.6.2.1.1

Factor $10$ out of $10 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.2.1.2

Rewrite $10$ as $1$ plus $9$

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (1 + 9) x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 1 x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.5.1.4.6.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 2 \pm \sqrt{4 ((- x^{2} + x) + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 2 \pm \sqrt{4 (x (- x + 1) - 9 (- x + 1))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{- 2 \pm \sqrt{4 ((- x + 1) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

$y = \frac{- 2 \pm \sqrt{4 (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.7

Rewrite $4 (- x + 1) (x - 9)$ as $2^{2} ((- x + 1^{2}) (x - 9))$ .

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Step 1.5.1.4.6.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.7.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.7.3

Add parentheses.

$y = \frac{- 2 \pm \sqrt{2^{2} ((- x + 1^{2}) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.1.9

One to any power is one.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.2

Multiply $2$ by $- 1$ .

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ .

$y = 1 \pm \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.6.4

Change the $\pm$ to $+$ .

$y = 1 + \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.5.1.4.7.1

Simplify the numerator.

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Step 1.5.1.4.7.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.4

Simplify.

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Step 1.5.1.4.7.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.4.2

Multiply $10$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.4.3

Multiply $4$ by $- 10$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x - 40}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.5

Subtract $40$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6

Rewrite $- 4 x^{2} + 40 x - 36$ in a factored form.

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Step 1.5.1.4.7.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x - 36$ .

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Step 1.5.1.4.7.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.1.2

Factor $4$ out of $40 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.1.3

Factor $4$ out of $- 36$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (- 9)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.2

Factor by grouping.

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Step 1.5.1.4.7.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 10$ .

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Step 1.5.1.4.7.1.6.2.1.1

Factor $10$ out of $10 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.2.1.2

Rewrite $10$ as $1$ plus $9$

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (1 + 9) x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 1 x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.5.1.4.7.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 2 \pm \sqrt{4 ((- x^{2} + x) + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 2 \pm \sqrt{4 (x (- x + 1) - 9 (- x + 1))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{- 2 \pm \sqrt{4 ((- x + 1) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

$y = \frac{- 2 \pm \sqrt{4 (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.7

Rewrite $4 (- x + 1) (x - 9)$ as $2^{2} ((- x + 1^{2}) (x - 9))$ .

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Step 1.5.1.4.7.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.7.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.7.3

Add parentheses.

$y = \frac{- 2 \pm \sqrt{2^{2} ((- x + 1^{2}) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.1.9

One to any power is one.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.2

Multiply $2$ by $- 1$ .

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ .

$y = 1 \pm \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.7.4

Change the $\pm$ to $-$ .

$y = 1 - \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.1.4.8

Consolidate the solutions.

$y = 1 + \sqrt{(- x + 1) (x - 9)}, 1 - \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.5.2

Find the intersection of $y = 1 + \sqrt{(- x + 1) (x - 9)}, 1 - \sqrt{(- x + 1) (x - 9)}$ and $x = 5$ .

No solution and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6

Solve $- x + 5 > \sqrt{(y + 3) (- y + 5)}$ when $- 3 \leq x < 5$ .

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Step 1.6.1

Solve $- x + 5 > \sqrt{(y + 3) (- y + 5)}$ for $y$ .

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Step 1.6.1.1

Rewrite so $y$ is on the left side of the inequality.

$\sqrt{(y + 3) (- y + 5)} < - x + 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{(y + 3) (- y + 5)}}^{2} < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3

Simplify each side of the inequality.

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Step 1.6.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(y + 3) (- y + 5)}$ as ${((y + 3) (- y + 5))}^{\frac{1}{2}}$ .

${({((y + 3) (- y + 5))}^{\frac{1}{2}})}^{2} < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2

Simplify the left side.

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Step 1.6.1.3.2.1

Simplify ${({((y + 3) (- y + 5))}^{\frac{1}{2}})}^{2}$ .

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Step 1.6.1.3.2.1.1

Multiply the exponents in ${({((y + 3) (- y + 5))}^{\frac{1}{2}})}^{2}$ .

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Step 1.6.1.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${((y + 3) (- y + 5))}^{\frac{1}{2} \cdot 2} < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.6.1.3.2.1.1.2.1

Cancel the common factor.

${((y + 3) (- y + 5))}^{\frac{1}{2} \cdot 2} < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.1.2.2

Rewrite the expression.

$(y + 3) (- y + 5) < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.2

Expand $(y + 3) (- y + 5)$ using the FOIL Method.

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Step 1.6.1.3.2.1.2.1

Apply the distributive property.

$y (- y + 5) + 3 (- y + 5) < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.2.2

Apply the distributive property.

$y (- y) + y \cdot 5 + 3 (- y + 5) < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.2.3

Apply the distributive property.

$y (- y) + y \cdot 5 + 3 (- y) + 3 \cdot 5 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.3

Simplify and combine like terms.

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Step 1.6.1.3.2.1.3.1

Simplify each term.

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Step 1.6.1.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$- y \cdot y + y \cdot 5 + 3 (- y) + 3 \cdot 5 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 1.6.1.3.2.1.3.1.2.1

Move $y$ .

$- (y \cdot y) + y \cdot 5 + 3 (- y) + 3 \cdot 5 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.3.1.2.2

Multiply $y$ by $y$ .

$- y^{2} + y \cdot 5 + 3 (- y) + 3 \cdot 5 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.3.1.3

Move $5$ to the left of $y$ .

$- y^{2} + 5 \cdot y + 3 (- y) + 3 \cdot 5 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.3.1.4

Multiply $- 1$ by $3$ .

$- y^{2} + 5 y - 3 y + 3 \cdot 5 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.3.1.5

Multiply $3$ by $5$ .

$- y^{2} + 5 y - 3 y + 15 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.3.2

Subtract $3 y$ from $5 y$ .

$- y^{2} + 2 y + 15 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.2.1.4

Simplify.

$- y^{2} + 2 y + 15 < {(- x + 5)}^{2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3

Simplify the right side.

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Step 1.6.1.3.3.1

Simplify ${(- x + 5)}^{2}$ .

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Step 1.6.1.3.3.1.1

Rewrite ${(- x + 5)}^{2}$ as $(- x + 5) (- x + 5)$ .

$- y^{2} + 2 y + 15 < (- x + 5) (- x + 5)$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.2

Expand $(- x + 5) (- x + 5)$ using the FOIL Method.

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Step 1.6.1.3.3.1.2.1

Apply the distributive property.

$- y^{2} + 2 y + 15 < - x (- x + 5) + 5 (- x + 5)$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.2.2

Apply the distributive property.

$- y^{2} + 2 y + 15 < - x (- x) - x \cdot 5 + 5 (- x + 5)$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.2.3

Apply the distributive property.

$- y^{2} + 2 y + 15 < - x (- x) - x \cdot 5 + 5 (- x) + 5 \cdot 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3

Simplify and combine like terms.

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Step 1.6.1.3.3.1.3.1

Simplify each term.

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Step 1.6.1.3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$- y^{2} + 2 y + 15 < - 1 \cdot (- 1 x \cdot x) - x \cdot 5 + 5 (- x) + 5 \cdot 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.6.1.3.3.1.3.1.2.1

Move $x$ .

$- y^{2} + 2 y + 15 < - 1 \cdot (- 1 (x \cdot x)) - x \cdot 5 + 5 (- x) + 5 \cdot 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$- y^{2} + 2 y + 15 < - 1 \cdot (- 1 x^{2}) - x \cdot 5 + 5 (- x) + 5 \cdot 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

$- y^{2} + 2 y + 15 < 1 x^{2} - x \cdot 5 + 5 (- x) + 5 \cdot 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

$- y^{2} + 2 y + 15 < x^{2} - x \cdot 5 + 5 (- x) + 5 \cdot 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3.1.5

Multiply $5$ by $- 1$ .

$- y^{2} + 2 y + 15 < x^{2} - 5 x + 5 (- x) + 5 \cdot 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3.1.6

Multiply $- 1$ by $5$ .

$- y^{2} + 2 y + 15 < x^{2} - 5 x - 5 x + 5 \cdot 5$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3.1.7

Multiply $5$ by $5$ .

$- y^{2} + 2 y + 15 < x^{2} - 5 x - 5 x + 25$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.3.3.1.3.2

Subtract $5 x$ from $- 5 x$ .

$- y^{2} + 2 y + 15 < x^{2} - 10 x + 25$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4

Solve for $y$ .

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Step 1.6.1.4.1

Move all terms to the left side of the equation and simplify.

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Step 1.6.1.4.1.1

Move all the expressions to the left side of the equation.

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Step 1.6.1.4.1.1.1

Subtract $x^{2}$ from both sides of the inequality.

$- y^{2} + 2 y + 15 - x^{2} < - 10 x + 25$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.1.1.2

Add $10 x$ to both sides of the inequality.

$- y^{2} + 2 y + 15 - x^{2} + 10 x < 25$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.1.1.3

Subtract $25$ from both sides of the inequality.

$- y^{2} + 2 y + 15 - x^{2} + 10 x - 25 < 0$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.1.2

Subtract $25$ from $15$ .

$- y^{2} + 2 y - x^{2} + 10 x - 10 < 0$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.2

Convert the inequality to an equation.

$- y^{2} + 2 y - x^{2} + 10 x - 10 = 0$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.4

Substitute the values $a = - 1$ , $b = 2$ , and $c = - x^{2} + 10 x - 10$ into the quadratic formula and solve for $y$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 1 \cdot (- x^{2} + 10 x - 10))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5

Simplify.

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Step 1.6.1.4.5.1

Simplify the numerator.

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Step 1.6.1.4.5.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.4

Simplify.

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Step 1.6.1.4.5.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.4.2

Multiply $10$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.4.3

Multiply $4$ by $- 10$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x - 40}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.5

Subtract $40$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6

Rewrite $- 4 x^{2} + 40 x - 36$ in a factored form.

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Step 1.6.1.4.5.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x - 36$ .

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Step 1.6.1.4.5.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.1.2

Factor $4$ out of $40 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.1.3

Factor $4$ out of $- 36$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (- 9)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.2

Factor by grouping.

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Step 1.6.1.4.5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 10$ .

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Step 1.6.1.4.5.1.6.2.1.1

Factor $10$ out of $10 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.2.1.2

Rewrite $10$ as $1$ plus $9$

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (1 + 9) x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 1 x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.6.1.4.5.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 2 \pm \sqrt{4 ((- x^{2} + x) + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 2 \pm \sqrt{4 (x (- x + 1) - 9 (- x + 1))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{- 2 \pm \sqrt{4 ((- x + 1) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

$y = \frac{- 2 \pm \sqrt{4 (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.7

Rewrite $4 (- x + 1) (x - 9)$ as $2^{2} ((- x + 1^{2}) (x - 9))$ .

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Step 1.6.1.4.5.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.7.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.7.3

Add parentheses.

$y = \frac{- 2 \pm \sqrt{2^{2} ((- x + 1^{2}) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.1.9

One to any power is one.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.2

Multiply $2$ by $- 1$ .

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ .

$y = 1 \pm \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.6.1.4.6.1

Simplify the numerator.

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Step 1.6.1.4.6.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.4

Simplify.

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Step 1.6.1.4.6.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.4.2

Multiply $10$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.4.3

Multiply $4$ by $- 10$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x - 40}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.5

Subtract $40$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6

Rewrite $- 4 x^{2} + 40 x - 36$ in a factored form.

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Step 1.6.1.4.6.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x - 36$ .

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Step 1.6.1.4.6.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.1.2

Factor $4$ out of $40 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.1.3

Factor $4$ out of $- 36$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (- 9)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.2

Factor by grouping.

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Step 1.6.1.4.6.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 10$ .

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Step 1.6.1.4.6.1.6.2.1.1

Factor $10$ out of $10 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.2.1.2

Rewrite $10$ as $1$ plus $9$

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (1 + 9) x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 1 x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.6.1.4.6.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 2 \pm \sqrt{4 ((- x^{2} + x) + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 2 \pm \sqrt{4 (x (- x + 1) - 9 (- x + 1))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{- 2 \pm \sqrt{4 ((- x + 1) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

$y = \frac{- 2 \pm \sqrt{4 (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.7

Rewrite $4 (- x + 1) (x - 9)$ as $2^{2} ((- x + 1^{2}) (x - 9))$ .

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Step 1.6.1.4.6.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.7.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.7.3

Add parentheses.

$y = \frac{- 2 \pm \sqrt{2^{2} ((- x + 1^{2}) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.1.9

One to any power is one.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.2

Multiply $2$ by $- 1$ .

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ .

$y = 1 \pm \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.6.4

Change the $\pm$ to $+$ .

$y = 1 + \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.6.1.4.7.1

Simplify the numerator.

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Step 1.6.1.4.7.1.1

Raise $2$ to the power of $2$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x - 10)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.4

Simplify.

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Step 1.6.1.4.7.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.4.2

Multiply $10$ by $4$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot - 10}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.4.3

Multiply $4$ by $- 10$ .

$y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x - 40}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.5

Subtract $40$ from $4$ .

$y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6

Rewrite $- 4 x^{2} + 40 x - 36$ in a factored form.

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Step 1.6.1.4.7.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x - 36$ .

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Step 1.6.1.4.7.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.1.2

Factor $4$ out of $40 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) - 36}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.1.3

Factor $4$ out of $- 36$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (- 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (- 9)$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.2

Factor by grouping.

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Step 1.6.1.4.7.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 9 = 9$ and whose sum is $b = 10$ .

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Step 1.6.1.4.7.1.6.2.1.1

Factor $10$ out of $10 x$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.2.1.2

Rewrite $10$ as $1$ plus $9$

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (1 + 9) x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 1 x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.2.1.4

Multiply $x$ by $1$ .

$y = \frac{- 2 \pm \sqrt{4 (- x^{2} + x + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.6.1.4.7.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 2 \pm \sqrt{4 ((- x^{2} + x) + 9 x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 2 \pm \sqrt{4 (x (- x + 1) - 9 (- x + 1))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$y = \frac{- 2 \pm \sqrt{4 ((- x + 1) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

$y = \frac{- 2 \pm \sqrt{4 (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.7

Rewrite $4 (- x + 1) (x - 9)$ as $2^{2} ((- x + 1^{2}) (x - 9))$ .

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Step 1.6.1.4.7.1.7.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.7.2

Rewrite $1$ as $1^{2}$ .

$y = \frac{- 2 \pm \sqrt{2^{2} (- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.7.3

Add parentheses.

$y = \frac{- 2 \pm \sqrt{2^{2} ((- x + 1^{2}) (x - 9))}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.8

Pull terms out from under the radical.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1^{2}) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.1.9

One to any power is one.

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{2 \cdot - 1}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.2

Multiply $2$ by $- 1$ .

$y = \frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x + 1) (x - 9)}}{- 2}$ .

$y = 1 \pm \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.7.4

Change the $\pm$ to $-$ .

$y = 1 - \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.1.4.8

Consolidate the solutions.

$y = 1 + \sqrt{(- x + 1) (x - 9)}, 1 - \sqrt{(- x + 1) (x - 9)}$ and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.6.2

Find the intersection of $y = 1 + \sqrt{(- x + 1) (x - 9)}, 1 - \sqrt{(- x + 1) (x - 9)}$ and $- 3 \leq x < 5$ .

No solution and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

Step 1.7

Find the union of the solutions.

No solution and ${(x - 5)}^{2} + {(y - 1)}^{2} < 36$

No solution

Step 2

Simplify the second inequality.

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Step 2.1

Subtract ${(y - 1)}^{2}$ from both sides of the inequality.

No solution and ${(x - 5)}^{2} < 36 - {(y - 1)}^{2}$

Step 2.2

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

No solution and $\sqrt{{(x - 5)}^{2}} < \sqrt{36 - {(y - 1)}^{2}}$

Step 2.3

Simplify the equation.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Pull terms out from under the radical.

No solution and $| x - 5 | < \sqrt{36 - {(y - 1)}^{2}}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{36 - {(y - 1)}^{2}}$ .

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Step 2.3.2.1.1

Rewrite $36$ as $6^{2}$ .

No solution and $| x - 5 | < \sqrt{6^{2} - {(y - 1)}^{2}}$

Step 2.3.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = y - 1$ .

No solution and $| x - 5 | < \sqrt{(6 + y - 1) (6 - (y - 1))}$

Step 2.3.2.1.3

Simplify.

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Step 2.3.2.1.3.1

Subtract $1$ from $6$ .

No solution and $| x - 5 | < \sqrt{(y + 5) (6 - (y - 1))}$

Step 2.3.2.1.3.2

Apply the distributive property.

No solution and $| x - 5 | < \sqrt{(y + 5) (6 - y + 1)}$

Step 2.3.2.1.3.3

Multiply $- 1$ by $- 1$ .

No solution and $| x - 5 | < \sqrt{(y + 5) (6 - y + 1)}$

Step 2.3.2.1.3.4

Add $6$ and $1$ .

No solution and $| x - 5 | < \sqrt{(y + 5) (- y + 7)}$

Step 2.4

Write $| x - 5 | < \sqrt{(y + 5) (- y + 7)}$ as a piecewise.

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Step 2.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x - 5 \geq 0$

Step 2.4.2

Add $5$ to both sides of the inequality.

$x \geq 5$

Step 2.4.3

In the piece where $x - 5$ is non-negative, remove the absolute value.

$x - 5 < \sqrt{(y + 5) (- y + 7)}$

Step 2.4.4

Find the domain of $x - 5 < \sqrt{(y + 5) (- y + 7)}$ and find the intersection with $x \geq 5$ .

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Step 2.4.4.1

Find the domain of $x - 5 < \sqrt{(y + 5) (- y + 7)}$ .

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Step 2.4.4.1.1

Set the radicand in $\sqrt{(y + 5) (- y + 7)}$ greater than or equal to $0$ to find where the expression is defined.

$(y + 5) (- y + 7) \geq 0$

Step 2.4.4.1.2

Solve for $x$ .

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Step 2.4.4.1.2.1

Simplify $(y + 5) (- y + 7)$ .

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Step 2.4.4.1.2.1.1

Expand $(y + 5) (- y + 7)$ using the FOIL Method.

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Step 2.4.4.1.2.1.1.1

Apply the distributive property.

$y (- y + 7) + 5 (- y + 7) \geq 0$

Step 2.4.4.1.2.1.1.2

Apply the distributive property.

$y (- y) + y \cdot 7 + 5 (- y + 7) \geq 0$

Step 2.4.4.1.2.1.1.3

Apply the distributive property.

$y (- y) + y \cdot 7 + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.4.1.2.1.2

Simplify and combine like terms.

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Step 2.4.4.1.2.1.2.1

Simplify each term.

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Step 2.4.4.1.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$- y \cdot y + y \cdot 7 + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.4.1.2.1.2.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.4.4.1.2.1.2.1.2.1

Move $y$ .

$- (y \cdot y) + y \cdot 7 + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.4.1.2.1.2.1.2.2

Multiply $y$ by $y$ .

$- y^{2} + y \cdot 7 + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.4.1.2.1.2.1.3

Move $7$ to the left of $y$ .

$- y^{2} + 7 \cdot y + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.4.1.2.1.2.1.4

Multiply $- 1$ by $5$ .

$- y^{2} + 7 y - 5 y + 5 \cdot 7 \geq 0$

Step 2.4.4.1.2.1.2.1.5

Multiply $5$ by $7$ .

$- y^{2} + 7 y - 5 y + 35 \geq 0$

Step 2.4.4.1.2.1.2.2

Subtract $5 y$ from $7 y$ .

$- y^{2} + 2 y + 35 \geq 0$

Step 2.4.4.1.2.2

Convert the inequality to an equation.

$- y^{2} + 2 y + 35 = 0$

Step 2.4.4.1.2.3

Factor the left side of the equation.

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Step 2.4.4.1.2.3.1

Factor $- 1$ out of $- y^{2} + 2 y + 35$ .

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Step 2.4.4.1.2.3.1.1

Factor $- 1$ out of $- y^{2}$ .

$- (y^{2}) + 2 y + 35 = 0$

Step 2.4.4.1.2.3.1.2

Factor $- 1$ out of $2 y$ .

$- (y^{2}) - (- 2 y) + 35 = 0$

Step 2.4.4.1.2.3.1.3

Rewrite $35$ as $- 1 (- 35)$ .

$- (y^{2}) - (- 2 y) - 1 \cdot - 35 = 0$

Step 2.4.4.1.2.3.1.4

Factor $- 1$ out of $- (y^{2}) - (- 2 y)$ .

$- (y^{2} - 2 y) - 1 \cdot - 35 = 0$

Step 2.4.4.1.2.3.1.5

Factor $- 1$ out of $- (y^{2} - 2 y) - 1 (- 35)$ .

$- (y^{2} - 2 y - 35) = 0$

Step 2.4.4.1.2.3.2

Factor.

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Step 2.4.4.1.2.3.2.1

Factor $y^{2} - 2 y - 35$ using the AC method.

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Step 2.4.4.1.2.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 35$ and whose sum is $- 2$ .

$- 7, 5$

Step 2.4.4.1.2.3.2.1.2

Write the factored form using these integers.

$- ((y - 7) (y + 5)) = 0$

Step 2.4.4.1.2.3.2.2

Remove unnecessary parentheses.

$- (y - 7) (y + 5) = 0$

Step 2.4.4.1.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 7 = 0$

$y + 5 = 0$

Step 2.4.4.1.2.5

Set $y - 7$ equal to $0$ and solve for $y$ .

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Step 2.4.4.1.2.5.1

Set $y - 7$ equal to $0$ .

$y - 7 = 0$

Step 2.4.4.1.2.5.2

Add $7$ to both sides of the equation.

$y = 7$

Step 2.4.4.1.2.6

Set $y + 5$ equal to $0$ and solve for $y$ .

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Step 2.4.4.1.2.6.1

Set $y + 5$ equal to $0$ .

$y + 5 = 0$

Step 2.4.4.1.2.6.2

Subtract $5$ from both sides of the equation.

$y = - 5$

Step 2.4.4.1.2.7

The final solution is all the values that make $- (y - 7) (y + 5) = 0$ true.

$y = 7, - 5$

Step 2.4.4.1.2.8

Use each root to create test intervals.

$y < - 5$

$- 5 < y < 7$

$y > 7$

Step 2.4.4.1.2.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 2.4.4.1.2.9.1

Test a value on the interval $y < - 5$ to see if it makes the inequality true.

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Step 2.4.4.1.2.9.1.1

Choose a value on the interval $y < - 5$ and see if this value makes the original inequality true.

$y = - 8$

Step 2.4.4.1.2.9.1.2

Replace $y$ with $- 8$ in the original inequality.

$((- 8) + 5) (- (- 8) + 7) \geq 0$

Step 2.4.4.1.2.9.1.3

The left side $- 45$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.4.4.1.2.9.2

Test a value on the interval $- 5 < y < 7$ to see if it makes the inequality true.

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Step 2.4.4.1.2.9.2.1

Choose a value on the interval $- 5 < y < 7$ and see if this value makes the original inequality true.

$y = 0$

Step 2.4.4.1.2.9.2.2

Replace $y$ with $0$ in the original inequality.

$((0) + 5) (- (0) + 7) \geq 0$

Step 2.4.4.1.2.9.2.3

The left side $35$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.4.4.1.2.9.3

Test a value on the interval $y > 7$ to see if it makes the inequality true.

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Step 2.4.4.1.2.9.3.1

Choose a value on the interval $y > 7$ and see if this value makes the original inequality true.

$y = 10$

Step 2.4.4.1.2.9.3.2

Replace $y$ with $10$ in the original inequality.

$((10) + 5) (- (10) + 7) \geq 0$

Step 2.4.4.1.2.9.3.3

The left side $- 45$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.4.4.1.2.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$y < - 5$ False

$- 5 < y < 7$ True

$y > 7$ False

$y < - 5$ False

$- 5 < y < 7$ True

$y > 7$ False

Step 2.4.4.1.2.10

The solution consists of all of the true intervals.

$- 5 \leq y \leq 7$

Step 2.4.4.1.3

The domain is all values of $x$ that make the expression defined.

$[- 5, 7]$

Step 2.4.4.2

Find the intersection of $x \geq 5$ and $[- 5, 7]$ .

$5 \leq x \leq 7$

Step 2.4.5

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x - 5 < 0$

Step 2.4.6

Add $5$ to both sides of the inequality.

$x < 5$

Step 2.4.7

In the piece where $x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 5) < \sqrt{(y + 5) (- y + 7)}$

Step 2.4.8

Find the domain of $- (x - 5) < \sqrt{(y + 5) (- y + 7)}$ and find the intersection with $x < 5$ .

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Step 2.4.8.1

Find the domain of $- (x - 5) < \sqrt{(y + 5) (- y + 7)}$ .

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Step 2.4.8.1.1

Set the radicand in $\sqrt{(y + 5) (- y + 7)}$ greater than or equal to $0$ to find where the expression is defined.

$(y + 5) (- y + 7) \geq 0$

Step 2.4.8.1.2

Solve for $x$ .

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Step 2.4.8.1.2.1

Simplify $(y + 5) (- y + 7)$ .

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Step 2.4.8.1.2.1.1

Expand $(y + 5) (- y + 7)$ using the FOIL Method.

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Step 2.4.8.1.2.1.1.1

Apply the distributive property.

$y (- y + 7) + 5 (- y + 7) \geq 0$

Step 2.4.8.1.2.1.1.2

Apply the distributive property.

$y (- y) + y \cdot 7 + 5 (- y + 7) \geq 0$

Step 2.4.8.1.2.1.1.3

Apply the distributive property.

$y (- y) + y \cdot 7 + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.8.1.2.1.2

Simplify and combine like terms.

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Step 2.4.8.1.2.1.2.1

Simplify each term.

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Step 2.4.8.1.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$- y \cdot y + y \cdot 7 + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.8.1.2.1.2.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.4.8.1.2.1.2.1.2.1

Move $y$ .

$- (y \cdot y) + y \cdot 7 + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.8.1.2.1.2.1.2.2

Multiply $y$ by $y$ .

$- y^{2} + y \cdot 7 + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.8.1.2.1.2.1.3

Move $7$ to the left of $y$ .

$- y^{2} + 7 \cdot y + 5 (- y) + 5 \cdot 7 \geq 0$

Step 2.4.8.1.2.1.2.1.4

Multiply $- 1$ by $5$ .

$- y^{2} + 7 y - 5 y + 5 \cdot 7 \geq 0$

Step 2.4.8.1.2.1.2.1.5

Multiply $5$ by $7$ .

$- y^{2} + 7 y - 5 y + 35 \geq 0$

Step 2.4.8.1.2.1.2.2

Subtract $5 y$ from $7 y$ .

$- y^{2} + 2 y + 35 \geq 0$

Step 2.4.8.1.2.2

Convert the inequality to an equation.

$- y^{2} + 2 y + 35 = 0$

Step 2.4.8.1.2.3

Factor the left side of the equation.

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Step 2.4.8.1.2.3.1

Factor $- 1$ out of $- y^{2} + 2 y + 35$ .

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Step 2.4.8.1.2.3.1.1

Factor $- 1$ out of $- y^{2}$ .

$- (y^{2}) + 2 y + 35 = 0$

Step 2.4.8.1.2.3.1.2

Factor $- 1$ out of $2 y$ .

$- (y^{2}) - (- 2 y) + 35 = 0$

Step 2.4.8.1.2.3.1.3

Rewrite $35$ as $- 1 (- 35)$ .

$- (y^{2}) - (- 2 y) - 1 \cdot - 35 = 0$

Step 2.4.8.1.2.3.1.4

Factor $- 1$ out of $- (y^{2}) - (- 2 y)$ .

$- (y^{2} - 2 y) - 1 \cdot - 35 = 0$

Step 2.4.8.1.2.3.1.5

Factor $- 1$ out of $- (y^{2} - 2 y) - 1 (- 35)$ .

$- (y^{2} - 2 y - 35) = 0$

Step 2.4.8.1.2.3.2

Factor.

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Step 2.4.8.1.2.3.2.1

Factor $y^{2} - 2 y - 35$ using the AC method.

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Step 2.4.8.1.2.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 35$ and whose sum is $- 2$ .

$- 7, 5$

Step 2.4.8.1.2.3.2.1.2

Write the factored form using these integers.

$- ((y - 7) (y + 5)) = 0$

Step 2.4.8.1.2.3.2.2

Remove unnecessary parentheses.

$- (y - 7) (y + 5) = 0$

Step 2.4.8.1.2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 7 = 0$

$y + 5 = 0$

Step 2.4.8.1.2.5

Set $y - 7$ equal to $0$ and solve for $y$ .

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Step 2.4.8.1.2.5.1

Set $y - 7$ equal to $0$ .

$y - 7 = 0$

Step 2.4.8.1.2.5.2

Add $7$ to both sides of the equation.

$y = 7$

Step 2.4.8.1.2.6

Set $y + 5$ equal to $0$ and solve for $y$ .

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Step 2.4.8.1.2.6.1

Set $y + 5$ equal to $0$ .

$y + 5 = 0$

Step 2.4.8.1.2.6.2

Subtract $5$ from both sides of the equation.

$y = - 5$

Step 2.4.8.1.2.7

The final solution is all the values that make $- (y - 7) (y + 5) = 0$ true.

$y = 7, - 5$

Step 2.4.8.1.2.8

Use each root to create test intervals.

$y < - 5$

$- 5 < y < 7$

$y > 7$

Step 2.4.8.1.2.9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 2.4.8.1.2.9.1

Test a value on the interval $y < - 5$ to see if it makes the inequality true.

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Step 2.4.8.1.2.9.1.1

Choose a value on the interval $y < - 5$ and see if this value makes the original inequality true.

$y = - 8$

Step 2.4.8.1.2.9.1.2

Replace $y$ with $- 8$ in the original inequality.

$((- 8) + 5) (- (- 8) + 7) \geq 0$

Step 2.4.8.1.2.9.1.3

The left side $- 45$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.4.8.1.2.9.2

Test a value on the interval $- 5 < y < 7$ to see if it makes the inequality true.

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Step 2.4.8.1.2.9.2.1

Choose a value on the interval $- 5 < y < 7$ and see if this value makes the original inequality true.

$y = 0$

Step 2.4.8.1.2.9.2.2

Replace $y$ with $0$ in the original inequality.

$((0) + 5) (- (0) + 7) \geq 0$

Step 2.4.8.1.2.9.2.3

The left side $35$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.4.8.1.2.9.3

Test a value on the interval $y > 7$ to see if it makes the inequality true.

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Step 2.4.8.1.2.9.3.1

Choose a value on the interval $y > 7$ and see if this value makes the original inequality true.

$y = 10$

Step 2.4.8.1.2.9.3.2

Replace $y$ with $10$ in the original inequality.

$((10) + 5) (- (10) + 7) \geq 0$

Step 2.4.8.1.2.9.3.3

The left side $- 45$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.4.8.1.2.9.4

Compare the intervals to determine which ones satisfy the original inequality.

$y < - 5$ False

$- 5 < y < 7$ True

$y > 7$ False

$y < - 5$ False

$- 5 < y < 7$ True

$y > 7$ False

Step 2.4.8.1.2.10

The solution consists of all of the true intervals.

$- 5 \leq y \leq 7$

Step 2.4.8.1.3

The domain is all values of $x$ that make the expression defined.

$[- 5, 7]$

Step 2.4.8.2

Find the intersection of $x < 5$ and $[- 5, 7]$ .

$- 5 \leq x < 5$

Step 2.4.9

Write as a piecewise.

No solution and ${\begin{cases} x - 5 < \sqrt{(y + 5) (- y + 7)} & 5 \leq x \leq 7 \\ - (x - 5) < \sqrt{(y + 5) (- y + 7)} & - 5 \leq x < 5 \end{cases}$

Step 2.4.10

Simplify $- (x - 5) < \sqrt{(y + 5) (- y + 7)}$ .

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Step 2.4.10.1

Apply the distributive property.

No solution and ${\begin{cases} x - 5 < \sqrt{(y + 5) (- y + 7)} & 5 \leq x \leq 7 \\ - x + 5 < \sqrt{(y + 5) (- y + 7)} & - 5 \leq x < 5 \end{cases}$

Step 2.4.10.2

Multiply $- 1$ by $- 5$ .

No solution and ${\begin{cases} x - 5 < \sqrt{(y + 5) (- y + 7)} & 5 \leq x \leq 7 \\ - x + 5 < \sqrt{(y + 5) (- y + 7)} & - 5 \leq x < 5 \end{cases}$

Step 2.5

Solve $x - 5 < \sqrt{(y + 5) (- y + 7)}$ when $5 \leq x \leq 7$ .

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Step 2.5.1

Solve $x - 5 < \sqrt{(y + 5) (- y + 7)}$ for $y$ .

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Step 2.5.1.1

Rewrite so $y$ is on the left side of the inequality.

No solution and $\sqrt{(y + 5) (- y + 7)} > x - 5$

Step 2.5.1.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

No solution and ${\sqrt{(y + 5) (- y + 7)}}^{2} > {(x - 5)}^{2}$

Step 2.5.1.3

Simplify each side of the inequality.

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Step 2.5.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(y + 5) (- y + 7)}$ as ${((y + 5) (- y + 7))}^{\frac{1}{2}}$ .

No solution and ${({((y + 5) (- y + 7))}^{\frac{1}{2}})}^{2} > {(x - 5)}^{2}$

Step 2.5.1.3.2

Simplify the left side.

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Step 2.5.1.3.2.1

Simplify ${({((y + 5) (- y + 7))}^{\frac{1}{2}})}^{2}$ .

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Step 2.5.1.3.2.1.1

Multiply the exponents in ${({((y + 5) (- y + 7))}^{\frac{1}{2}})}^{2}$ .

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Step 2.5.1.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

No solution and ${((y + 5) (- y + 7))}^{\frac{1}{2} \cdot 2} > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.5.1.3.2.1.1.2.1

Cancel the common factor.

No solution and ${((y + 5) (- y + 7))}^{\frac{1}{2} \cdot 2} > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.1.2.2

Rewrite the expression.

No solution and $(y + 5) (- y + 7) > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.2

Expand $(y + 5) (- y + 7)$ using the FOIL Method.

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Step 2.5.1.3.2.1.2.1

Apply the distributive property.

No solution and $y (- y + 7) + 5 (- y + 7) > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.2.2

Apply the distributive property.

No solution and $y (- y) + y \cdot 7 + 5 (- y + 7) > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.2.3

Apply the distributive property.

No solution and $y (- y) + y \cdot 7 + 5 (- y) + 5 \cdot 7 > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.3

Simplify and combine like terms.

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Step 2.5.1.3.2.1.3.1

Simplify each term.

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Step 2.5.1.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

No solution and $- y \cdot y + y \cdot 7 + 5 (- y) + 5 \cdot 7 > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.5.1.3.2.1.3.1.2.1

Move $y$ .

No solution and $- (y \cdot y) + y \cdot 7 + 5 (- y) + 5 \cdot 7 > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.3.1.2.2

Multiply $y$ by $y$ .

No solution and $- y^{2} + y \cdot 7 + 5 (- y) + 5 \cdot 7 > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.3.1.3

Move $7$ to the left of $y$ .

No solution and $- y^{2} + 7 \cdot y + 5 (- y) + 5 \cdot 7 > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.3.1.4

Multiply $- 1$ by $5$ .

No solution and $- y^{2} + 7 y - 5 y + 5 \cdot 7 > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.3.1.5

Multiply $5$ by $7$ .

No solution and $- y^{2} + 7 y - 5 y + 35 > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.3.2

Subtract $5 y$ from $7 y$ .

No solution and $- y^{2} + 2 y + 35 > {(x - 5)}^{2}$

Step 2.5.1.3.2.1.4

Simplify.

No solution and $- y^{2} + 2 y + 35 > {(x - 5)}^{2}$

Step 2.5.1.3.3

Simplify the right side.

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Step 2.5.1.3.3.1

Simplify ${(x - 5)}^{2}$ .

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Step 2.5.1.3.3.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

No solution and $- y^{2} + 2 y + 35 > (x - 5) (x - 5)$

Step 2.5.1.3.3.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 2.5.1.3.3.1.2.1

Apply the distributive property.

No solution and $- y^{2} + 2 y + 35 > x (x - 5) - 5 (x - 5)$

Step 2.5.1.3.3.1.2.2

Apply the distributive property.

No solution and $- y^{2} + 2 y + 35 > x \cdot x + x \cdot - 5 - 5 (x - 5)$

Step 2.5.1.3.3.1.2.3

Apply the distributive property.

No solution and $- y^{2} + 2 y + 35 > x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5$

Step 2.5.1.3.3.1.3

Simplify and combine like terms.

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Step 2.5.1.3.3.1.3.1

Simplify each term.

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Step 2.5.1.3.3.1.3.1.1

Multiply $x$ by $x$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5$

Step 2.5.1.3.3.1.3.1.2

Move $- 5$ to the left of $x$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5$

Step 2.5.1.3.3.1.3.1.3

Multiply $- 5$ by $- 5$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} - 5 x - 5 x + 25$

Step 2.5.1.3.3.1.3.2

Subtract $5 x$ from $- 5 x$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} - 10 x + 25$

Step 2.5.1.4

Solve for $y$ .

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Step 2.5.1.4.1

Move all terms to the left side of the equation and simplify.

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Step 2.5.1.4.1.1

Move all the expressions to the left side of the equation.

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Step 2.5.1.4.1.1.1

Subtract $x^{2}$ from both sides of the inequality.

No solution and $- y^{2} + 2 y + 35 - x^{2} > - 10 x + 25$

Step 2.5.1.4.1.1.2

Add $10 x$ to both sides of the inequality.

No solution and $- y^{2} + 2 y + 35 - x^{2} + 10 x > 25$

Step 2.5.1.4.1.1.3

Subtract $25$ from both sides of the inequality.

No solution and $- y^{2} + 2 y + 35 - x^{2} + 10 x - 25 > 0$

Step 2.5.1.4.1.2

Subtract $25$ from $35$ .

No solution and $- y^{2} + 2 y - x^{2} + 10 x + 10 > 0$

Step 2.5.1.4.2

Convert the inequality to an equation.

No solution and $- y^{2} + 2 y - x^{2} + 10 x + 10 = 0$

Step 2.5.1.4.3

Use the quadratic formula to find the solutions.

No solution and $\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5.1.4.4

Substitute the values $a = - 1$ , $b = 2$ , and $c = - x^{2} + 10 x + 10$ into the quadratic formula and solve for $y$ .

No solution and $\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 1 \cdot (- x^{2} + 10 x + 10))}}{2 \cdot - 1}$

Step 2.5.1.4.5

Simplify.

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Step 2.5.1.4.5.1

Simplify the numerator.

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Step 2.5.1.4.5.1.1

Raise $2$ to the power of $2$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.2

Multiply $- 4$ by $- 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.4

Simplify.

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Step 2.5.1.4.5.1.4.1

Multiply $- 1$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.4.2

Multiply $10$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.4.3

Multiply $4$ by $10$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 40}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.5

Add $4$ and $40$ .

No solution and $y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x + 44}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6

Rewrite $- 4 x^{2} + 40 x + 44$ in a factored form.

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Step 2.5.1.4.5.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x + 44$ .

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Step 2.5.1.4.5.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x + 44}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.1.2

Factor $4$ out of $40 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 44}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.1.3

Factor $4$ out of $44$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (11)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.2

Factor by grouping.

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Step 2.5.1.4.5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 11 = - 11$ and whose sum is $b = 10$ .

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Step 2.5.1.4.5.1.6.2.1.1

Factor $10$ out of $10 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.2.1.2

Rewrite $10$ as $- 1$ plus $11$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (- 1 + 11) x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.2.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} - 1 x + 11 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 2.5.1.4.5.1.6.2.2.1

Group the first two terms and the last two terms.

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x^{2} - 1 x) + 11 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

No solution and $y = \frac{- 2 \pm \sqrt{4 (x (- x - 1) - 11 (- x - 1))}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x - 1) (x - 11))}}{2 \cdot - 1}$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.7

Rewrite $4 (- x - 1) (x - 11)$ as $2^{2} ((- x - 1) (x - 11))$ .

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Step 2.5.1.4.5.1.7.1

Rewrite $4$ as $2^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.7.2

Add parentheses.

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} ((- x - 1) (x - 11))}}{2 \cdot - 1}$

Step 2.5.1.4.5.1.8

Pull terms out from under the radical.

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.5.2

Multiply $2$ by $- 1$ .

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$

Step 2.5.1.4.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$ .

No solution and $y = 1 \pm \sqrt{(- x - 1) (x - 11)}$

Step 2.5.1.4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.5.1.4.6.1

Simplify the numerator.

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Step 2.5.1.4.6.1.1

Raise $2$ to the power of $2$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.2

Multiply $- 4$ by $- 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.4

Simplify.

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Step 2.5.1.4.6.1.4.1

Multiply $- 1$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.4.2

Multiply $10$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.4.3

Multiply $4$ by $10$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 40}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.5

Add $4$ and $40$ .

No solution and $y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x + 44}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6

Rewrite $- 4 x^{2} + 40 x + 44$ in a factored form.

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Step 2.5.1.4.6.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x + 44$ .

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Step 2.5.1.4.6.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x + 44}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.1.2

Factor $4$ out of $40 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 44}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.1.3

Factor $4$ out of $44$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (11)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.2

Factor by grouping.

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Step 2.5.1.4.6.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 11 = - 11$ and whose sum is $b = 10$ .

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Step 2.5.1.4.6.1.6.2.1.1

Factor $10$ out of $10 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.2.1.2

Rewrite $10$ as $- 1$ plus $11$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (- 1 + 11) x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.2.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} - 1 x + 11 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.2.2

Factor out the greatest common factor from each group.

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Step 2.5.1.4.6.1.6.2.2.1

Group the first two terms and the last two terms.

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x^{2} - 1 x) + 11 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

No solution and $y = \frac{- 2 \pm \sqrt{4 (x (- x - 1) - 11 (- x - 1))}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x - 1) (x - 11))}}{2 \cdot - 1}$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.7

Rewrite $4 (- x - 1) (x - 11)$ as $2^{2} ((- x - 1) (x - 11))$ .

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Step 2.5.1.4.6.1.7.1

Rewrite $4$ as $2^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.7.2

Add parentheses.

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} ((- x - 1) (x - 11))}}{2 \cdot - 1}$

Step 2.5.1.4.6.1.8

Pull terms out from under the radical.

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.6.2

Multiply $2$ by $- 1$ .

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$

Step 2.5.1.4.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$ .

No solution and $y = 1 \pm \sqrt{(- x - 1) (x - 11)}$

Step 2.5.1.4.6.4

Change the $\pm$ to $+$ .

No solution and $y = 1 + \sqrt{(- x - 1) (x - 11)}$

Step 2.5.1.4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.1.4.7.1

Simplify the numerator.

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Step 2.5.1.4.7.1.1

Raise $2$ to the power of $2$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.2

Multiply $- 4$ by $- 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.4

Simplify.

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Step 2.5.1.4.7.1.4.1

Multiply $- 1$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.4.2

Multiply $10$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.4.3

Multiply $4$ by $10$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 40}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.5

Add $4$ and $40$ .

No solution and $y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x + 44}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6

Rewrite $- 4 x^{2} + 40 x + 44$ in a factored form.

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Step 2.5.1.4.7.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x + 44$ .

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Step 2.5.1.4.7.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x + 44}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.1.2

Factor $4$ out of $40 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 44}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.1.3

Factor $4$ out of $44$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (11)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.2

Factor by grouping.

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Step 2.5.1.4.7.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 11 = - 11$ and whose sum is $b = 10$ .

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Step 2.5.1.4.7.1.6.2.1.1

Factor $10$ out of $10 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.2.1.2

Rewrite $10$ as $- 1$ plus $11$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (- 1 + 11) x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.2.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} - 1 x + 11 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.2.2

Factor out the greatest common factor from each group.

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Step 2.5.1.4.7.1.6.2.2.1

Group the first two terms and the last two terms.

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x^{2} - 1 x) + 11 x + 11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

No solution and $y = \frac{- 2 \pm \sqrt{4 (x (- x - 1) - 11 (- x - 1))}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x - 1) (x - 11))}}{2 \cdot - 1}$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.7

Rewrite $4 (- x - 1) (x - 11)$ as $2^{2} ((- x - 1) (x - 11))$ .

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Step 2.5.1.4.7.1.7.1

Rewrite $4$ as $2^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.7.2

Add parentheses.

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} ((- x - 1) (x - 11))}}{2 \cdot - 1}$

Step 2.5.1.4.7.1.8

Pull terms out from under the radical.

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.5.1.4.7.2

Multiply $2$ by $- 1$ .

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$

Step 2.5.1.4.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$ .

No solution and $y = 1 \pm \sqrt{(- x - 1) (x - 11)}$

Step 2.5.1.4.7.4

Change the $\pm$ to $-$ .

No solution and $y = 1 - \sqrt{(- x - 1) (x - 11)}$

Step 2.5.1.4.8

Consolidate the solutions.

No solution and $y = 1 + \sqrt{(- x - 1) (x - 11)}, 1 - \sqrt{(- x - 1) (x - 11)}$

Step 2.5.2

Find the intersection of $y = 1 + \sqrt{(- x - 1) (x - 11)}, 1 - \sqrt{(- x - 1) (x - 11)}$ and $5 \leq x \leq 7$ .

No solution and No solution

Step 2.6

Solve $- x + 5 < \sqrt{(y + 5) (- y + 7)}$ when $- 5 \leq x < 5$ .

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Step 2.6.1

Solve $- x + 5 < \sqrt{(y + 5) (- y + 7)}$ for $y$ .

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Step 2.6.1.1

Rewrite so $y$ is on the left side of the inequality.

No solution and $\sqrt{(y + 5) (- y + 7)} > - x + 5$

Step 2.6.1.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

No solution and ${\sqrt{(y + 5) (- y + 7)}}^{2} > {(- x + 5)}^{2}$

Step 2.6.1.3

Simplify each side of the inequality.

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Step 2.6.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(y + 5) (- y + 7)}$ as ${((y + 5) (- y + 7))}^{\frac{1}{2}}$ .

No solution and ${({((y + 5) (- y + 7))}^{\frac{1}{2}})}^{2} > {(- x + 5)}^{2}$

Step 2.6.1.3.2

Simplify the left side.

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Step 2.6.1.3.2.1

Simplify ${({((y + 5) (- y + 7))}^{\frac{1}{2}})}^{2}$ .

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Step 2.6.1.3.2.1.1

Multiply the exponents in ${({((y + 5) (- y + 7))}^{\frac{1}{2}})}^{2}$ .

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Step 2.6.1.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

No solution and ${((y + 5) (- y + 7))}^{\frac{1}{2} \cdot 2} > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.6.1.3.2.1.1.2.1

Cancel the common factor.

No solution and ${((y + 5) (- y + 7))}^{\frac{1}{2} \cdot 2} > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.1.2.2

Rewrite the expression.

No solution and $(y + 5) (- y + 7) > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.2

Expand $(y + 5) (- y + 7)$ using the FOIL Method.

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Step 2.6.1.3.2.1.2.1

Apply the distributive property.

No solution and $y (- y + 7) + 5 (- y + 7) > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.2.2

Apply the distributive property.

No solution and $y (- y) + y \cdot 7 + 5 (- y + 7) > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.2.3

Apply the distributive property.

No solution and $y (- y) + y \cdot 7 + 5 (- y) + 5 \cdot 7 > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.3

Simplify and combine like terms.

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Step 2.6.1.3.2.1.3.1

Simplify each term.

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Step 2.6.1.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

No solution and $- y \cdot y + y \cdot 7 + 5 (- y) + 5 \cdot 7 > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.6.1.3.2.1.3.1.2.1

Move $y$ .

No solution and $- (y \cdot y) + y \cdot 7 + 5 (- y) + 5 \cdot 7 > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.3.1.2.2

Multiply $y$ by $y$ .

No solution and $- y^{2} + y \cdot 7 + 5 (- y) + 5 \cdot 7 > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.3.1.3

Move $7$ to the left of $y$ .

No solution and $- y^{2} + 7 \cdot y + 5 (- y) + 5 \cdot 7 > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.3.1.4

Multiply $- 1$ by $5$ .

No solution and $- y^{2} + 7 y - 5 y + 5 \cdot 7 > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.3.1.5

Multiply $5$ by $7$ .

No solution and $- y^{2} + 7 y - 5 y + 35 > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.3.2

Subtract $5 y$ from $7 y$ .

No solution and $- y^{2} + 2 y + 35 > {(- x + 5)}^{2}$

Step 2.6.1.3.2.1.4

Simplify.

No solution and $- y^{2} + 2 y + 35 > {(- x + 5)}^{2}$

Step 2.6.1.3.3

Simplify the right side.

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Step 2.6.1.3.3.1

Simplify ${(- x + 5)}^{2}$ .

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Step 2.6.1.3.3.1.1

Rewrite ${(- x + 5)}^{2}$ as $(- x + 5) (- x + 5)$ .

No solution and $- y^{2} + 2 y + 35 > (- x + 5) (- x + 5)$

Step 2.6.1.3.3.1.2

Expand $(- x + 5) (- x + 5)$ using the FOIL Method.

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Step 2.6.1.3.3.1.2.1

Apply the distributive property.

No solution and $- y^{2} + 2 y + 35 > - x (- x + 5) + 5 (- x + 5)$

Step 2.6.1.3.3.1.2.2

Apply the distributive property.

No solution and $- y^{2} + 2 y + 35 > - x (- x) - x \cdot 5 + 5 (- x + 5)$

Step 2.6.1.3.3.1.2.3

Apply the distributive property.

No solution and $- y^{2} + 2 y + 35 > - x (- x) - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 2.6.1.3.3.1.3

Simplify and combine like terms.

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Step 2.6.1.3.3.1.3.1

Simplify each term.

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Step 2.6.1.3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

No solution and $- y^{2} + 2 y + 35 > - 1 \cdot (- 1 x \cdot x) - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 2.6.1.3.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.6.1.3.3.1.3.1.2.1

Move $x$ .

No solution and $- y^{2} + 2 y + 35 > - 1 \cdot (- 1 (x \cdot x)) - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 2.6.1.3.3.1.3.1.2.2

Multiply $x$ by $x$ .

No solution and $- y^{2} + 2 y + 35 > - 1 \cdot (- 1 x^{2}) - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 2.6.1.3.3.1.3.1.3

Multiply $- 1$ by $- 1$ .

No solution and $- y^{2} + 2 y + 35 > 1 x^{2} - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 2.6.1.3.3.1.3.1.4

Multiply $x^{2}$ by $1$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} - x \cdot 5 + 5 (- x) + 5 \cdot 5$

Step 2.6.1.3.3.1.3.1.5

Multiply $5$ by $- 1$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} - 5 x + 5 (- x) + 5 \cdot 5$

Step 2.6.1.3.3.1.3.1.6

Multiply $- 1$ by $5$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} - 5 x - 5 x + 5 \cdot 5$

Step 2.6.1.3.3.1.3.1.7

Multiply $5$ by $5$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} - 5 x - 5 x + 25$

Step 2.6.1.3.3.1.3.2

Subtract $5 x$ from $- 5 x$ .

No solution and $- y^{2} + 2 y + 35 > x^{2} - 10 x + 25$

Step 2.6.1.4

Solve for $y$ .

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Step 2.6.1.4.1

Move all terms to the left side of the equation and simplify.

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Step 2.6.1.4.1.1

Move all the expressions to the left side of the equation.

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Step 2.6.1.4.1.1.1

Subtract $x^{2}$ from both sides of the inequality.

No solution and $- y^{2} + 2 y + 35 - x^{2} > - 10 x + 25$

Step 2.6.1.4.1.1.2

Add $10 x$ to both sides of the inequality.

No solution and $- y^{2} + 2 y + 35 - x^{2} + 10 x > 25$

Step 2.6.1.4.1.1.3

Subtract $25$ from both sides of the inequality.

No solution and $- y^{2} + 2 y + 35 - x^{2} + 10 x - 25 > 0$

Step 2.6.1.4.1.2

Subtract $25$ from $35$ .

No solution and $- y^{2} + 2 y - x^{2} + 10 x + 10 > 0$

Step 2.6.1.4.2

Convert the inequality to an equation.

No solution and $- y^{2} + 2 y - x^{2} + 10 x + 10 = 0$

Step 2.6.1.4.3

Use the quadratic formula to find the solutions.

No solution and $\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.6.1.4.4

Substitute the values $a = - 1$ , $b = 2$ , and $c = - x^{2} + 10 x + 10$ into the quadratic formula and solve for $y$ .

No solution and $\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 1 \cdot (- x^{2} + 10 x + 10))}}{2 \cdot - 1}$

Step 2.6.1.4.5

Simplify.

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Step 2.6.1.4.5.1

Simplify the numerator.

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Step 2.6.1.4.5.1.1

Raise $2$ to the power of $2$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.2

Multiply $- 4$ by $- 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.4

Simplify.

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Step 2.6.1.4.5.1.4.1

Multiply $- 1$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.4.2

Multiply $10$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.4.3

Multiply $4$ by $10$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 40}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.5

Add $4$ and $40$ .

No solution and $y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x + 44}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6

Rewrite $- 4 x^{2} + 40 x + 44$ in a factored form.

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Step 2.6.1.4.5.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x + 44$ .

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Step 2.6.1.4.5.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x + 44}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.1.2

Factor $4$ out of $40 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 44}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.1.3

Factor $4$ out of $44$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (11)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.2

Factor by grouping.

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Step 2.6.1.4.5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 11 = - 11$ and whose sum is $b = 10$ .

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Step 2.6.1.4.5.1.6.2.1.1

Factor $10$ out of $10 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.2.1.2

Rewrite $10$ as $- 1$ plus $11$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (- 1 + 11) x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.2.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} - 1 x + 11 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 2.6.1.4.5.1.6.2.2.1

Group the first two terms and the last two terms.

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x^{2} - 1 x) + 11 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

No solution and $y = \frac{- 2 \pm \sqrt{4 (x (- x - 1) - 11 (- x - 1))}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x - 1) (x - 11))}}{2 \cdot - 1}$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.7

Rewrite $4 (- x - 1) (x - 11)$ as $2^{2} ((- x - 1) (x - 11))$ .

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Step 2.6.1.4.5.1.7.1

Rewrite $4$ as $2^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.7.2

Add parentheses.

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} ((- x - 1) (x - 11))}}{2 \cdot - 1}$

Step 2.6.1.4.5.1.8

Pull terms out from under the radical.

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.5.2

Multiply $2$ by $- 1$ .

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$

Step 2.6.1.4.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$ .

No solution and $y = 1 \pm \sqrt{(- x - 1) (x - 11)}$

Step 2.6.1.4.6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.6.1.4.6.1

Simplify the numerator.

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Step 2.6.1.4.6.1.1

Raise $2$ to the power of $2$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.2

Multiply $- 4$ by $- 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.4

Simplify.

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Step 2.6.1.4.6.1.4.1

Multiply $- 1$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.4.2

Multiply $10$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.4.3

Multiply $4$ by $10$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 40}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.5

Add $4$ and $40$ .

No solution and $y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x + 44}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6

Rewrite $- 4 x^{2} + 40 x + 44$ in a factored form.

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Step 2.6.1.4.6.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x + 44$ .

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Step 2.6.1.4.6.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x + 44}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.1.2

Factor $4$ out of $40 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 44}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.1.3

Factor $4$ out of $44$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (11)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.2

Factor by grouping.

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Step 2.6.1.4.6.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 11 = - 11$ and whose sum is $b = 10$ .

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Step 2.6.1.4.6.1.6.2.1.1

Factor $10$ out of $10 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.2.1.2

Rewrite $10$ as $- 1$ plus $11$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (- 1 + 11) x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.2.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} - 1 x + 11 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.2.2

Factor out the greatest common factor from each group.

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Step 2.6.1.4.6.1.6.2.2.1

Group the first two terms and the last two terms.

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x^{2} - 1 x) + 11 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

No solution and $y = \frac{- 2 \pm \sqrt{4 (x (- x - 1) - 11 (- x - 1))}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x - 1) (x - 11))}}{2 \cdot - 1}$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.7

Rewrite $4 (- x - 1) (x - 11)$ as $2^{2} ((- x - 1) (x - 11))$ .

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Step 2.6.1.4.6.1.7.1

Rewrite $4$ as $2^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.7.2

Add parentheses.

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} ((- x - 1) (x - 11))}}{2 \cdot - 1}$

Step 2.6.1.4.6.1.8

Pull terms out from under the radical.

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.6.2

Multiply $2$ by $- 1$ .

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$

Step 2.6.1.4.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$ .

No solution and $y = 1 \pm \sqrt{(- x - 1) (x - 11)}$

Step 2.6.1.4.6.4

Change the $\pm$ to $+$ .

No solution and $y = 1 + \sqrt{(- x - 1) (x - 11)}$

Step 2.6.1.4.7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.6.1.4.7.1

Simplify the numerator.

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Step 2.6.1.4.7.1.1

Raise $2$ to the power of $2$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.2

Multiply $- 4$ by $- 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 \cdot (- x^{2} + 10 x + 10)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 + 4 (- x^{2}) + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.4

Simplify.

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Step 2.6.1.4.7.1.4.1

Multiply $- 1$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 4 (10 x) + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.4.2

Multiply $10$ by $4$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 4 \cdot 10}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.4.3

Multiply $4$ by $10$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 - 4 x^{2} + 40 x + 40}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.5

Add $4$ and $40$ .

No solution and $y = \frac{- 2 \pm \sqrt{- 4 x^{2} + 40 x + 44}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6

Rewrite $- 4 x^{2} + 40 x + 44$ in a factored form.

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Step 2.6.1.4.7.1.6.1

Factor $4$ out of $- 4 x^{2} + 40 x + 44$ .

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Step 2.6.1.4.7.1.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 40 x + 44}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.1.2

Factor $4$ out of $40 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 44}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.1.3

Factor $4$ out of $44$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2}) + 4 (10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (10 x)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x) + 4 (11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.1.5

Factor $4$ out of $4 (- x^{2} + 10 x) + 4 (11)$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.2

Factor by grouping.

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Step 2.6.1.4.7.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 11 = - 11$ and whose sum is $b = 10$ .

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Step 2.6.1.4.7.1.6.2.1.1

Factor $10$ out of $10 x$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + 10 (x) + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.2.1.2

Rewrite $10$ as $- 1$ plus $11$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} + (- 1 + 11) x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.2.1.3

Apply the distributive property.

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x^{2} - 1 x + 11 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.2.2

Factor out the greatest common factor from each group.

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Step 2.6.1.4.7.1.6.2.2.1

Group the first two terms and the last two terms.

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x^{2} - 1 x) + 11 x + 11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

No solution and $y = \frac{- 2 \pm \sqrt{4 (x (- x - 1) - 11 (- x - 1))}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

No solution and $y = \frac{- 2 \pm \sqrt{4 ((- x - 1) (x - 11))}}{2 \cdot - 1}$

No solution and $y = \frac{- 2 \pm \sqrt{4 (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.7

Rewrite $4 (- x - 1) (x - 11)$ as $2^{2} ((- x - 1) (x - 11))$ .

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Step 2.6.1.4.7.1.7.1

Rewrite $4$ as $2^{2}$ .

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} (- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.7.2

Add parentheses.

No solution and $y = \frac{- 2 \pm \sqrt{2^{2} ((- x - 1) (x - 11))}}{2 \cdot - 1}$

Step 2.6.1.4.7.1.8

Pull terms out from under the radical.

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{2 \cdot - 1}$

Step 2.6.1.4.7.2

Multiply $2$ by $- 1$ .

No solution and $y = \frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$

Step 2.6.1.4.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{(- x - 1) (x - 11)}}{- 2}$ .

No solution and $y = 1 \pm \sqrt{(- x - 1) (x - 11)}$

Step 2.6.1.4.7.4

Change the $\pm$ to $-$ .

No solution and $y = 1 - \sqrt{(- x - 1) (x - 11)}$

Step 2.6.1.4.8

Consolidate the solutions.

No solution and $y = 1 + \sqrt{(- x - 1) (x - 11)}, 1 - \sqrt{(- x - 1) (x - 11)}$

Step 2.6.2

Find the intersection of $y = 1 + \sqrt{(- x - 1) (x - 11)}, 1 - \sqrt{(- x - 1) (x - 11)}$ and $- 5 \leq x < 5$ .

No solution and No solution

Step 2.7

Find the union of the solutions.

No solution and No solution

No solution