Algebra Examples

$y \cdot y < - 2 x - 5$

Step 1

Solve for $y$ .

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Step 1.1

Multiply $y$ by $y$ .

$y^{2} < - 2 x - 5$

Step 1.2

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

$\sqrt{y^{2}} < \sqrt{- 2 x - 5}$

Step 1.3

Simplify the left side.

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Step 1.3.1

Pull terms out from under the radical.

$| y | < \sqrt{- 2 x - 5}$

Step 1.4

Write $| y | < \sqrt{- 2 x - 5}$ 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.

$y \geq 0$

Step 1.4.2

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

$y < \sqrt{- 2 x - 5}$

Step 1.4.3

Find the domain of $y < \sqrt{- 2 x - 5}$ and find the intersection with $y \geq 0$ .

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Step 1.4.3.1

Find the domain of $y < \sqrt{- 2 x - 5}$ .

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Step 1.4.3.1.1

Set the radicand in $\sqrt{- 2 x - 5}$ greater than or equal to $0$ to find where the expression is defined.

$- 2 x - 5 \geq 0$

Step 1.4.3.1.2

Solve for $y$ .

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Step 1.4.3.1.2.1

Add $5$ to both sides of the inequality.

$- 2 x \geq 5$

Step 1.4.3.1.2.2

Divide each term in $- 2 x \geq 5$ by $- 2$ and simplify.

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Step 1.4.3.1.2.2.1

Divide each term in $- 2 x \geq 5$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{5}{- 2}$

Step 1.4.3.1.2.2.2

Simplify the left side.

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Step 1.4.3.1.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.4.3.1.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{5}{- 2}$

Step 1.4.3.1.2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{5}{- 2}$

Step 1.4.3.1.2.2.3

Simplify the right side.

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Step 1.4.3.1.2.2.3.1

Move the negative in front of the fraction.

$x \leq - \frac{5}{2}$

Step 1.4.3.1.3

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

$(- \infty, - \frac{5}{2}]$

Step 1.4.3.2

Find the intersection of $y \geq 0$ and $(- \infty, - \frac{5}{2}]$ .

$No solution$

Step 1.4.4

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

$y < 0$

Step 1.4.5

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

$- y < \sqrt{- 2 x - 5}$

Step 1.4.6

Find the domain of $- y < \sqrt{- 2 x - 5}$ and find the intersection with $y < 0$ .

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Step 1.4.6.1

Find the domain of $- y < \sqrt{- 2 x - 5}$ .

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Step 1.4.6.1.1

Set the radicand in $\sqrt{- 2 x - 5}$ greater than or equal to $0$ to find where the expression is defined.

$- 2 x - 5 \geq 0$

Step 1.4.6.1.2

Solve for $y$ .

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Step 1.4.6.1.2.1

Add $5$ to both sides of the inequality.

$- 2 x \geq 5$

Step 1.4.6.1.2.2

Divide each term in $- 2 x \geq 5$ by $- 2$ and simplify.

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Step 1.4.6.1.2.2.1

Divide each term in $- 2 x \geq 5$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{5}{- 2}$

Step 1.4.6.1.2.2.2

Simplify the left side.

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Step 1.4.6.1.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 1.4.6.1.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{5}{- 2}$

Step 1.4.6.1.2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{5}{- 2}$

Step 1.4.6.1.2.2.3

Simplify the right side.

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Step 1.4.6.1.2.2.3.1

Move the negative in front of the fraction.

$x \leq - \frac{5}{2}$

Step 1.4.6.1.3

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

$(- \infty, - \frac{5}{2}]$

Step 1.4.6.2

Find the intersection of $y < 0$ and $(- \infty, - \frac{5}{2}]$ .

$y \leq - \frac{5}{2}$

Step 1.4.7

Write as a piecewise.

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

Step 1.5

Solve $- y < \sqrt{- 2 x - 5}$ when $y \leq - \frac{5}{2}$ .

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Step 1.5.1

Divide each term in $- y < \sqrt{- 2 x - 5}$ by $- 1$ and simplify.

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Step 1.5.1.1

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

$\frac{- y}{- 1} > \frac{\sqrt{- 2 x - 5}}{- 1}$

Step 1.5.1.2

Simplify the left side.

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Step 1.5.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} > \frac{\sqrt{- 2 x - 5}}{- 1}$

Step 1.5.1.2.2

Divide $y$ by $1$ .

$y > \frac{\sqrt{- 2 x - 5}}{- 1}$

Step 1.5.1.3

Simplify the right side.

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Step 1.5.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{- 2 x - 5}}{- 1}$ .

$y > - 1 \cdot \sqrt{- 2 x - 5}$

Step 1.5.1.3.2

Rewrite $- 1 \cdot \sqrt{- 2 x - 5}$ as $- \sqrt{- 2 x - 5}$ .

$y > - \sqrt{- 2 x - 5}$

Step 1.5.2

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

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

Step 1.6

Find the union of the solutions.

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

Step 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 .

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

Step 4