Algebra Examples

$\sqrt{x + 3} > \sqrt{9 - x^{2}}$

Step 1

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

${\sqrt{x + 3}}^{2} > {\sqrt{9 - x^{2}}}^{2}$

Step 2

Simplify each side of the inequality.

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Step 2.1

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

${({(x + 3)}^{\frac{1}{2}})}^{2} > {\sqrt{9 - x^{2}}}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

${(x + 3)}^{\frac{1}{2} \cdot 2} > {\sqrt{9 - x^{2}}}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x + 3)}^{\frac{1}{2} \cdot 2} > {\sqrt{9 - x^{2}}}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x + 3)}^{1} > {\sqrt{9 - x^{2}}}^{2}$

Step 2.2.1.2

Simplify.

$x + 3 > {\sqrt{9 - x^{2}}}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${\sqrt{9 - x^{2}}}^{2}$ .

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Step 2.3.1.1

Rewrite $9$ as $3^{2}$ .

$x + 3 > {\sqrt{3^{2} - x^{2}}}^{2}$

Step 2.3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$x + 3 > {\sqrt{(3 + x) (3 - x)}}^{2}$

Step 2.3.1.3

Rewrite ${\sqrt{(3 + x) (3 - x)}}^{2}$ as $(3 + x) (3 - x)$ .

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Step 2.3.1.3.1

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

$x + 3 > {({((3 + x) (3 - x))}^{\frac{1}{2}})}^{2}$

Step 2.3.1.3.2

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

$x + 3 > {((3 + x) (3 - x))}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$x + 3 > {((3 + x) (3 - x))}^{\frac{2}{2}}$

Step 2.3.1.3.4

Cancel the common factor of $2$ .

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Step 2.3.1.3.4.1

Cancel the common factor.

$x + 3 > {((3 + x) (3 - x))}^{\frac{2}{2}}$

Step 2.3.1.3.4.2

Rewrite the expression.

$x + 3 > {((3 + x) (3 - x))}^{1}$

Step 2.3.1.3.5

Simplify.

$x + 3 > (3 + x) (3 - x)$

Step 2.3.1.4

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

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Step 2.3.1.4.1

Apply the distributive property.

$x + 3 > 3 (3 - x) + x (3 - x)$

Step 2.3.1.4.2

Apply the distributive property.

$x + 3 > 3 \cdot 3 + 3 (- x) + x (3 - x)$

Step 2.3.1.4.3

Apply the distributive property.

$x + 3 > 3 \cdot 3 + 3 (- x) + x \cdot 3 + x (- x)$

Step 2.3.1.5

Simplify and combine like terms.

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Step 2.3.1.5.1

Simplify each term.

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Step 2.3.1.5.1.1

Multiply $3$ by $3$ .

$x + 3 > 9 + 3 (- x) + x \cdot 3 + x (- x)$

Step 2.3.1.5.1.2

Multiply $- 1$ by $3$ .

$x + 3 > 9 - 3 x + x \cdot 3 + x (- x)$

Step 2.3.1.5.1.3

Move $3$ to the left of $x$ .

$x + 3 > 9 - 3 x + 3 \cdot x + x (- x)$

Step 2.3.1.5.1.4

Rewrite using the commutative property of multiplication.

$x + 3 > 9 - 3 x + 3 x - x \cdot x$

Step 2.3.1.5.1.5

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

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Step 2.3.1.5.1.5.1

Move $x$ .

$x + 3 > 9 - 3 x + 3 x - (x \cdot x)$

Step 2.3.1.5.1.5.2

Multiply $x$ by $x$ .

$x + 3 > 9 - 3 x + 3 x - x^{2}$

Step 2.3.1.5.2

Add $- 3 x$ and $3 x$ .

$x + 3 > 9 + 0 - x^{2}$

Step 2.3.1.5.3

Add $9$ and $0$ .

$x + 3 > 9 - x^{2}$

Step 3

Solve for $x$ .

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Step 3.1

Add $x^{2}$ to both sides of the inequality.

$x + 3 + x^{2} > 9$

Step 3.2

Convert the inequality to an equation.

$x + 3 + x^{2} = 9$

Step 3.3

Subtract $9$ from both sides of the equation.

$x + 3 + x^{2} - 9 = 0$

Step 3.4

Subtract $9$ from $3$ .

$x + x^{2} - 6 = 0$

Step 3.5

Factor the left side of the equation.

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Step 3.5.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$u + u^{2} - 6 = 0$

Step 3.5.2

Factor $u + u^{2} - 6$ using the AC method.

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Step 3.5.2.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 3.5.2.2

Write the factored form using these integers.

$(u - 2) (u + 3) = 0$

Step 3.5.3

Replace all occurrences of $u$ with $x$ .

$(x - 2) (x + 3) = 0$

Step 3.6

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

$x - 2 = 0$

$x + 3 = 0$

Step 3.7

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.7.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.8

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

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Step 3.8.1

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

$x + 3 = 0$

Step 3.8.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.9

The final solution is all the values that make $(x - 2) (x + 3) = 0$ true.

$x = 2, - 3$

Step 4

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

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Step 4.1

Set the radicand in $\sqrt{x + 3}$ greater than or equal to $0$ to find where the expression is defined.

$x + 3 \geq 0$

Step 4.2

Subtract $3$ from both sides of the inequality.

$x \geq - 3$

Step 4.3

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

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

Step 4.4

Solve for $x$ .

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Step 4.4.1

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

$3 + x = 0$

$3 - x = 0$

Step 4.4.2

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

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Step 4.4.2.1

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

$3 + x = 0$

Step 4.4.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.4.3

Set $3 - x$ equal to $0$ and solve for $x$ .

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Step 4.4.3.1

Set $3 - x$ equal to $0$ .

$3 - x = 0$

Step 4.4.3.2

Solve $3 - x = 0$ for $x$ .

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Step 4.4.3.2.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 4.4.3.2.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 4.4.3.2.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 4.4.3.2.2.2

Simplify the left side.

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Step 4.4.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 4.4.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 4.4.3.2.2.3

Simplify the right side.

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Step 4.4.3.2.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 4.4.4

The final solution is all the values that make $(3 + x) (3 - x) \geq 0$ true.

$x = - 3, 3$

Step 4.4.5

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 3$

$x > 3$

Step 4.4.6

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 4.4.6.1

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

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Step 4.4.6.1.1

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

$x = - 6$

Step 4.4.6.1.2

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

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

Step 4.4.6.1.3

The left side $- 27$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.4.6.2

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

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Step 4.4.6.2.1

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

$x = 0$

Step 4.4.6.2.2

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

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

Step 4.4.6.2.3

The left side $9$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 4.4.6.3

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 4.4.6.3.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 4.4.6.3.2

Replace $x$ with $6$ in the original inequality.

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

Step 4.4.6.3.3

The left side $- 27$ is less than the right side $0$ , which means that the given statement is false.

False

Step 4.4.6.4

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

$x < - 3$ False

$- 3 < x < 3$ True

$x > 3$ False

$x < - 3$ False

$- 3 < x < 3$ True

$x > 3$ False

Step 4.4.7

The solution consists of all of the true intervals.

$- 3 \leq x \leq 3$

Step 4.5

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

$[- 3, 3]$

Step 5

Use each root to create test intervals.

$x < - 3$

$- 3 < x < 2$

$2 < x < 3$

$x > 3$

Step 6

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 6.1

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

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Step 6.1.1

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

$x = - 6$

Step 6.1.2

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

$\sqrt{(- 6) + 3} > \sqrt{9 - {(- 6)}^{2}}$

Step 6.1.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 6.2

Test a value on the interval $- 3 < x < 2$ to see if it makes the inequality true.

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Step 6.2.1

Choose a value on the interval $- 3 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 6.2.2

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

$\sqrt{(0) + 3} > \sqrt{9 - {(0)}^{2}}$

Step 6.2.3

The left side $1.7320508$ is not greater than the right side $3$ , which means that the given statement is false.

False

Step 6.3

Test a value on the interval $2 < x < 3$ to see if it makes the inequality true.

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Step 6.3.1

Choose a value on the interval $2 < x < 3$ and see if this value makes the original inequality true.

$x = 2.5$

Step 6.3.2

Replace $x$ with $2.5$ in the original inequality.

$\sqrt{(2.5) + 3} > \sqrt{9 - {(2.5)}^{2}}$

Step 6.3.3

The left side $2.34520787$ is greater than the right side $1.65831239$ , which means that the given statement is always true.

True

Step 6.4

Test a value on the interval $x > 3$ to see if it makes the inequality true.

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Step 6.4.1

Choose a value on the interval $x > 3$ and see if this value makes the original inequality true.

$x = 6$

Step 6.4.2

Replace $x$ with $6$ in the original inequality.

$\sqrt{(6) + 3} > \sqrt{9 - {(6)}^{2}}$

Step 6.4.3

The left side is not equal to the right side, which means that the given statement is false.

False

Step 6.5

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

$x < - 3$ False

$- 3 < x < 2$ False

$2 < x < 3$ True

$x > 3$ False

$x < - 3$ False

$- 3 < x < 2$ False

$2 < x < 3$ True

$x > 3$ False

Step 7

The solution consists of all of the true intervals.

$2 < x < 3$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$2 < x < 3$

Interval Notation:

$(2, 3)$

Step 9