Graph yy<=-2x-2

$y \cdot y \leq - 2 x - 2$

Solve for $y$ .

Multiply $y$ by $y$ .

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

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

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

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| y | \leq \sqrt{- 2 x - 2}$

Simplify the right side.

Factor $2$ out of $- 2 x - 2$ .

Factor $2$ out of $- 2 x$ .

$| y | \leq \sqrt{2 (- x) - 2}$

Factor $2$ out of $- 2$ .

$| y | \leq \sqrt{2 (- x) + 2 (- 1)}$

Factor $2$ out of $2 (- x) + 2 (- 1)$ .

$| y | \leq \sqrt{2 (- x - 1)}$

Write $| y | \leq \sqrt{2 (- x - 1)}$ 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.

$y \leq \sqrt{2 (- x - 1)}$

Find the domain of $y \leq \sqrt{2 (- x - 1)}$ and find the intersection with $y \geq 0$ .

Find the domain of $y \leq \sqrt{2 (- x - 1)}$ .

Set the radicand in $\sqrt{2 (- x - 1)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (- x - 1) \geq 0$

Solve for $y$ .

Divide each term in $2 (- x - 1) \geq 0$ by $2$ and simplify.

Divide each term in $2 (- x - 1) \geq 0$ by $2$ .

$\frac{2 (- x - 1)}{2} \geq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (- x - 1)}{2} \geq \frac{0}{2}$

Divide $- x - 1$ by $1$ .

$- x - 1 \geq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$- x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$- x \geq 1$

Divide each term in $- x \geq 1$ by $- 1$ and simplify.

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

$\frac{- x}{- 1} \leq \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{1}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$x \leq - 1$

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

$(- \infty, - 1]$

Find the intersection of $y \geq 0$ and $(- \infty, - 1]$ .

$No solution$

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 $- 1$ .

$- y \leq \sqrt{2 (- x - 1)}$

Find the domain of $- y \leq \sqrt{2 (- x - 1)}$ and find the intersection with $y < 0$ .

Find the domain of $- y \leq \sqrt{2 (- x - 1)}$ .

Set the radicand in $\sqrt{2 (- x - 1)}$ greater than or equal to $0$ to find where the expression is defined.

$2 (- x - 1) \geq 0$

Solve for $y$ .

Divide each term in $2 (- x - 1) \geq 0$ by $2$ and simplify.

Divide each term in $2 (- x - 1) \geq 0$ by $2$ .

$\frac{2 (- x - 1)}{2} \geq \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (- x - 1)}{2} \geq \frac{0}{2}$

Divide $- x - 1$ by $1$ .

$- x - 1 \geq \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$- x - 1 \geq 0$

Add $1$ to both sides of the inequality.

$- x \geq 1$

Divide each term in $- x \geq 1$ by $- 1$ and simplify.

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

$\frac{- x}{- 1} \leq \frac{1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{1}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{1}{- 1}$

Simplify the right side.

Divide $1$ by $- 1$ .

$x \leq - 1$

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

$(- \infty, - 1]$

Find the intersection of $y < 0$ and $(- \infty, - 1]$ .

$y \leq - 1$

Write as a piecewise.

${\begin{cases} - y \leq \sqrt{2 (- x - 1)} & y \leq - 1 \end{cases}$

Solve $- y \leq \sqrt{2 (- x - 1)}$ when $y \leq - 1$ .

Divide each term in $- y \leq \sqrt{2 (- x - 1)}$ by $- 1$ and simplify.

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

$\frac{- y}{- 1} \geq \frac{\sqrt{2 (- x - 1)}}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} \geq \frac{\sqrt{2 (- x - 1)}}{- 1}$

Divide $y$ by $1$ .

$y \geq \frac{\sqrt{2 (- x - 1)}}{- 1}$

Simplify the right side.

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

$y \geq - 1 \cdot \sqrt{2 (- x - 1)}$

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

$y \geq - \sqrt{2 (- x - 1)}$

Find the intersection of $y \geq - \sqrt{2 (- x - 1)}$ and $y \leq - 1$ .

$- \sqrt{2 (- x - 1)} \leq y \leq - 1$

Find the union of the solutions.

$- \sqrt{2 (- x - 1)} \leq y \leq - 1$

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

Not Linear

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

$- \sqrt{2 (- x - 1)} \leq y \leq - 1$

Graph yy<=-2x-2

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