Algebra Examples

$4 \leq - x^{2} - y$

Step 1

Solve for $y$ .

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Step 1.1

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

$- x^{2} - y \geq 4$

Step 1.2

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

$- y \geq 4 + x^{2}$

Step 1.3

Divide each term in $- y \geq 4 + x^{2}$ by $- 1$ and simplify.

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Step 1.3.1

Divide each term in $- y \geq 4 + x^{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \leq \frac{4}{- 1} + \frac{x^{2}}{- 1}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} \leq \frac{4}{- 1} + \frac{x^{2}}{- 1}$

Step 1.3.2.2

Divide $y$ by $1$ .

$y \leq \frac{4}{- 1} + \frac{x^{2}}{- 1}$

Step 1.3.3

Simplify the right side.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

Divide $4$ by $- 1$ .

$y \leq - 4 + \frac{x^{2}}{- 1}$

Step 1.3.3.1.2

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

$y \leq - 4 - 1 \cdot x^{2}$

Step 1.3.3.1.3

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

$y \leq - 4 - x^{2}$

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

Reorder $- 4$ and $- x^{2}$ .

$y \leq - x^{2} - 4$

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 $- 4 - x^{2}$ .

$y \leq - 4 - x^{2}$

Step 4