Algebra Examples

$x \geq - 3 {(y - 2)}^{2} - 5$

Step 1

Solve for $y$ .

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Step 1.1

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

$- 3 {(y - 2)}^{2} - 5 \leq x$

Step 1.2

Add $5$ to both sides of the inequality.

$- 3 {(y - 2)}^{2} \leq x + 5$

Step 2

Find the slope and the y-intercept for the boundary line.

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Step 2.1

Rewrite in slope-intercept form.

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Step 2.1.1

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

Step 2.1.2

Divide each term in $- 3 {(y - 2)}^{2} \leq x + 5$ by $- 3$ and simplify.

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Step 2.1.2.1

Divide each term in $- 3 {(y - 2)}^{2} \leq x + 5$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 {(y - 2)}^{2}}{- 3} \geq \frac{x}{- 3} + \frac{5}{- 3}$

Step 2.1.2.2

Simplify the left side.

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Step 2.1.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.1.2.2.1.1

Cancel the common factor.

$\frac{- 3 {(y - 2)}^{2}}{- 3} \geq \frac{x}{- 3} + \frac{5}{- 3}$

Step 2.1.2.2.1.2

Divide ${(y - 2)}^{2}$ by $1$ .

${(y - 2)}^{2} \geq \frac{x}{- 3} + \frac{5}{- 3}$

Step 2.1.2.3

Simplify the right side.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Move the negative in front of the fraction.

${(y - 2)}^{2} \geq - \frac{x}{3} + \frac{5}{- 3}$

Step 2.1.2.3.1.2

Move the negative in front of the fraction.

${(y - 2)}^{2} \geq - \frac{x}{3} - \frac{5}{3}$

Step 2.1.3

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

$\sqrt{{(y - 2)}^{2}} \geq \sqrt{- \frac{x}{3} - \frac{5}{3}}$

Step 2.1.4

Simplify the equation.

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Step 2.1.4.1

Simplify the left side.

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Step 2.1.4.1.1

Pull terms out from under the radical.

$| y - 2 | \geq \sqrt{- \frac{x}{3} - \frac{5}{3}}$

Step 2.1.4.2

Simplify the right side.

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Step 2.1.4.2.1

Simplify $\sqrt{- \frac{x}{3} - \frac{5}{3}}$ .

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Step 2.1.4.2.1.1

Factor $- 1$ out of $- \frac{x}{3} - \frac{5}{3}$ .

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Step 2.1.4.2.1.1.1

Factor $- 1$ out of $- \frac{x}{3}$ .

$| y - 2 | \geq \sqrt{- (\frac{x}{3}) - \frac{5}{3}}$

Step 2.1.4.2.1.1.2

Factor $- 1$ out of $- \frac{5}{3}$ .

$| y - 2 | \geq \sqrt{- (\frac{x}{3}) - (\frac{5}{3})}$

Step 2.1.4.2.1.1.3

Factor $- 1$ out of $- (\frac{x}{3}) - (\frac{5}{3})$ .

$| y - 2 | \geq \sqrt{- (\frac{x}{3} + \frac{5}{3})}$

Step 2.1.4.2.1.2

Combine the numerators over the common denominator.

$| y - 2 | \geq \sqrt{- \frac{x + 5}{3}}$

Step 2.1.5

Write $| y - 2 | \geq \sqrt{- \frac{x + 5}{3}}$ as a piecewise.

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Step 2.1.5.1

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

$y - 2 \geq 0$

Step 2.1.5.2

Add $2$ to both sides of the inequality.

$y \geq 2$

Step 2.1.5.3

In the piece where $y - 2$ is non-negative, remove the absolute value.

$y - 2 \geq \sqrt{- \frac{x + 5}{3}}$

Step 2.1.5.4

Find the domain of $y - 2 \geq \sqrt{- \frac{x + 5}{3}}$ and find the intersection with $y \geq 2$ .

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Step 2.1.5.4.1

Find the domain of $y - 2 \geq \sqrt{- \frac{x + 5}{3}}$ .

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Step 2.1.5.4.1.1

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

$- \frac{x + 5}{3} \geq 0$

Step 2.1.5.4.1.2

Solve for $y$ .

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Step 2.1.5.4.1.2.1

Divide each term in $- \frac{x + 5}{3} \geq 0$ by $- 1$ and simplify.

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Step 2.1.5.4.1.2.1.1

Divide each term in $- \frac{x + 5}{3} \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x + 5}{3}}{- 1} \leq \frac{0}{- 1}$

Step 2.1.5.4.1.2.1.2

Simplify the left side.

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Step 2.1.5.4.1.2.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x + 5}{3}}{1} \leq \frac{0}{- 1}$

Step 2.1.5.4.1.2.1.2.2

Divide $\frac{x + 5}{3}$ by $1$ .

$\frac{x + 5}{3} \leq \frac{0}{- 1}$

Step 2.1.5.4.1.2.1.3

Simplify the right side.

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Step 2.1.5.4.1.2.1.3.1

Divide $0$ by $- 1$ .

$\frac{x + 5}{3} \leq 0$

Step 2.1.5.4.1.2.2

Multiply both sides by $3$ .

$\frac{x + 5}{3} \cdot 3 \leq 0 \cdot 3$

Step 2.1.5.4.1.2.3

Simplify.

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Step 2.1.5.4.1.2.3.1

Simplify the left side.

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Step 2.1.5.4.1.2.3.1.1

Cancel the common factor of $3$ .

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Step 2.1.5.4.1.2.3.1.1.1

Cancel the common factor.

$\frac{x + 5}{3} \cdot 3 \leq 0 \cdot 3$

Step 2.1.5.4.1.2.3.1.1.2

Rewrite the expression.

$x + 5 \leq 0 \cdot 3$

Step 2.1.5.4.1.2.3.2

Simplify the right side.

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Step 2.1.5.4.1.2.3.2.1

Multiply $0$ by $3$ .

$x + 5 \leq 0$

Step 2.1.5.4.1.2.4

Subtract $5$ from both sides of the inequality.

$x \leq - 5$

Step 2.1.5.4.1.3

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

$(- \infty, - 5]$

Step 2.1.5.4.2

Find the intersection of $y \geq 2$ and $(- \infty, - 5]$ .

$No solution$

Step 2.1.5.5

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

$y - 2 < 0$

Step 2.1.5.6

Add $2$ to both sides of the inequality.

$y < 2$

Step 2.1.5.7

In the piece where $y - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (y - 2) \geq \sqrt{- \frac{x + 5}{3}}$

Step 2.1.5.8

Find the domain of $- (y - 2) \geq \sqrt{- \frac{x + 5}{3}}$ and find the intersection with $y < 2$ .

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Step 2.1.5.8.1

Find the domain of $- (y - 2) \geq \sqrt{- \frac{x + 5}{3}}$ .

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Step 2.1.5.8.1.1

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

$- \frac{x + 5}{3} \geq 0$

Step 2.1.5.8.1.2

Solve for $y$ .

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Step 2.1.5.8.1.2.1

Divide each term in $- \frac{x + 5}{3} \geq 0$ by $- 1$ and simplify.

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Step 2.1.5.8.1.2.1.1

Divide each term in $- \frac{x + 5}{3} \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x + 5}{3}}{- 1} \leq \frac{0}{- 1}$

Step 2.1.5.8.1.2.1.2

Simplify the left side.

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Step 2.1.5.8.1.2.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x + 5}{3}}{1} \leq \frac{0}{- 1}$

Step 2.1.5.8.1.2.1.2.2

Divide $\frac{x + 5}{3}$ by $1$ .

$\frac{x + 5}{3} \leq \frac{0}{- 1}$

Step 2.1.5.8.1.2.1.3

Simplify the right side.

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Step 2.1.5.8.1.2.1.3.1

Divide $0$ by $- 1$ .

$\frac{x + 5}{3} \leq 0$

Step 2.1.5.8.1.2.2

Multiply both sides by $3$ .

$\frac{x + 5}{3} \cdot 3 \leq 0 \cdot 3$

Step 2.1.5.8.1.2.3

Simplify.

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Step 2.1.5.8.1.2.3.1

Simplify the left side.

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Step 2.1.5.8.1.2.3.1.1

Cancel the common factor of $3$ .

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Step 2.1.5.8.1.2.3.1.1.1

Cancel the common factor.

$\frac{x + 5}{3} \cdot 3 \leq 0 \cdot 3$

Step 2.1.5.8.1.2.3.1.1.2

Rewrite the expression.

$x + 5 \leq 0 \cdot 3$

Step 2.1.5.8.1.2.3.2

Simplify the right side.

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Step 2.1.5.8.1.2.3.2.1

Multiply $0$ by $3$ .

$x + 5 \leq 0$

Step 2.1.5.8.1.2.4

Subtract $5$ from both sides of the inequality.

$x \leq - 5$

Step 2.1.5.8.1.3

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

$(- \infty, - 5]$

Step 2.1.5.8.2

Find the intersection of $y < 2$ and $(- \infty, - 5]$ .

$y \leq - 5$

Step 2.1.5.9

Write as a piecewise.

${\begin{cases} - (y - 2) \geq \sqrt{- \frac{x + 5}{3}} & y \leq - 5 \end{cases}$

Step 2.1.5.10

Simplify $- (y - 2) \geq \sqrt{- \frac{x + 5}{3}}$ .

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Step 2.1.5.10.1

Apply the distributive property.

${\begin{cases} - y - - 2 \geq \sqrt{- \frac{x + 5}{3}} & y \leq - 5 \end{cases}$

Step 2.1.5.10.2

Multiply $- 1$ by $- 2$ .

${\begin{cases} - y + 2 \geq \sqrt{- \frac{x + 5}{3}} & y \leq - 5 \end{cases}$

Step 2.1.6

Solve $- y + 2 \geq \sqrt{- \frac{x + 5}{3}}$ when $y \leq - 5$ .

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Step 2.1.6.1

Solve $- y + 2 \geq \sqrt{- \frac{x + 5}{3}}$ for $y$ .

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Step 2.1.6.1.1

Move all terms not containing $y$ to the right side of the inequality.

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Step 2.1.6.1.1.1

Subtract $2$ from both sides of the inequality.

$- y \geq \sqrt{- \frac{x + 5}{3}} - 2$

Step 2.1.6.1.1.2

Split the fraction $\frac{x + 5}{3}$ into two fractions.

$- y \geq \sqrt{- (\frac{x}{3} + \frac{5}{3})} - 2$

Step 2.1.6.1.2

Divide each term in $- y \geq \sqrt{- (\frac{x}{3} + \frac{5}{3})} - 2$ by $- 1$ and simplify.

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Step 2.1.6.1.2.1

Divide each term in $- y \geq \sqrt{- (\frac{x}{3} + \frac{5}{3})} - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \leq \frac{\sqrt{- (\frac{x}{3} + \frac{5}{3})}}{- 1} + \frac{- 2}{- 1}$

Step 2.1.6.1.2.2

Simplify the left side.

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Step 2.1.6.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} \leq \frac{\sqrt{- (\frac{x}{3} + \frac{5}{3})}}{- 1} + \frac{- 2}{- 1}$

Step 2.1.6.1.2.2.2

Divide $y$ by $1$ .

$y \leq \frac{\sqrt{- (\frac{x}{3} + \frac{5}{3})}}{- 1} + \frac{- 2}{- 1}$

Step 2.1.6.1.2.3

Simplify the right side.

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Step 2.1.6.1.2.3.1

Simplify each term.

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Step 2.1.6.1.2.3.1.1

Move the negative one from the denominator of $\frac{\sqrt{- (\frac{x}{3} + \frac{5}{3})}}{- 1}$ .

$y \leq - 1 \cdot \sqrt{- (\frac{x}{3} + \frac{5}{3})} + \frac{- 2}{- 1}$

Step 2.1.6.1.2.3.1.2

Rewrite $- 1 \cdot \sqrt{- (\frac{x}{3} + \frac{5}{3})}$ as $- \sqrt{- (\frac{x}{3} + \frac{5}{3})}$ .

$y \leq - \sqrt{- (\frac{x}{3} + \frac{5}{3})} + \frac{- 2}{- 1}$

Step 2.1.6.1.2.3.1.3

Combine the numerators over the common denominator.

$y \leq - \sqrt{- \frac{x + 5}{3}} + \frac{- 2}{- 1}$

Step 2.1.6.1.2.3.1.4

Divide $- 2$ by $- 1$ .

$y \leq - \sqrt{- \frac{x + 5}{3}} + 2$

Step 2.1.6.2

Find the intersection of $y \leq - \sqrt{- \frac{x + 5}{3}} + 2$ and $y \leq - 5$ .

$y \leq N o (Min i m u m)$

Step 2.1.7

Find the union of the solutions.

$y \leq N o (Min i m u m)$

Step 2.2

The equation is not linear, so a constant slope does not exist.

Not Linear

Step 3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $x + 5$ .

$- 3 {(y - 2)}^{2} \leq x + 5$

Step 4