Graph yy<=-3x-5

$y \cdot y \leq - 3 x - 5$

Step 1

Solve for $y$ .

Multiply $y$ by $y$ .

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

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

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

Simplify the left side.

Pull terms out from under the radical.

$| y | \leq \sqrt{- 3 x - 5}$

Write $| y | \leq \sqrt{- 3 x - 5}$ as a piecewise.

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

$y \geq 0$

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

$y \leq \sqrt{- 3 x - 5}$

Find the domain of $y \leq \sqrt{- 3 x - 5}$ and find the intersection with $y \geq 0$ .

Find the domain of $y \leq \sqrt{- 3 x - 5}$ .

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

$- 3 x - 5 \geq 0$

Solve for $y$ .

Add $5$ to both sides of the inequality.

$- 3 x \geq 5$

Divide each term in $- 3 x \geq 5$ by $- 3$ and simplify.

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

$\frac{- 3 x}{- 3} \leq \frac{5}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{5}{- 3}$

Divide $x$ by $1$ .

$x \leq \frac{5}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$x \leq - \frac{5}{3}$

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

$(- \infty, - \frac{5}{3}]$

Find the intersection of $y \geq 0$ and $(- \infty, - \frac{5}{3}]$ .

$No solution$

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

$y < 0$

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

$- y \leq \sqrt{- 3 x - 5}$

Find the domain of $- y \leq \sqrt{- 3 x - 5}$ and find the intersection with $y < 0$ .

Find the domain of $- y \leq \sqrt{- 3 x - 5}$ .

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

$- 3 x - 5 \geq 0$

Solve for $y$ .

Add $5$ to both sides of the inequality.

$- 3 x \geq 5$

Divide each term in $- 3 x \geq 5$ by $- 3$ and simplify.

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

$\frac{- 3 x}{- 3} \leq \frac{5}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{5}{- 3}$

Divide $x$ by $1$ .

$x \leq \frac{5}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$x \leq - \frac{5}{3}$

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

$(- \infty, - \frac{5}{3}]$

Find the intersection of $y < 0$ and $(- \infty, - \frac{5}{3}]$ .

$y \leq - \frac{5}{3}$

Write as a piecewise.

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

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

Divide each term in $- y \leq \sqrt{- 3 x - 5}$ by $- 1$ and simplify.

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

$\frac{- y}{- 1} \geq \frac{\sqrt{- 3 x - 5}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} \geq \frac{\sqrt{- 3 x - 5}}{- 1}$

Divide $y$ by $1$ .

$y \geq \frac{\sqrt{- 3 x - 5}}{- 1}$

Simplify the right side.

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

$y \geq - 1 \cdot \sqrt{- 3 x - 5}$

Rewrite $- 1 \cdot \sqrt{- 3 x - 5}$ as $- \sqrt{- 3 x - 5}$ .

$y \geq - \sqrt{- 3 x - 5}$

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

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

Find the union of the solutions.

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

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 .

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

Step 4

Graph yy<=-3x-5

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