Algebra Examples

$- y \geq x + 4$ $- 3 x + 3 y \leq - 9$

Step 1

Graph $- y \geq x + 4$ .

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Dividing two negative values results in a positive value.

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

Step 1.1.2.2

Divide $y$ by $1$ .

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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

Step 1.1.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

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

Step 1.1.3.1.3

Divide $4$ by $- 1$ .

$y \leq - x - 4$

Step 1.2

Use the slope-intercept form to find the slope and y-intercept.

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Step 1.2.1

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 1.2.2

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = - 1$

$b = - 4$

Step 1.2.3

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $- 1$

y-intercept: $(0, - 4)$

Slope: $- 1$

y-intercept: $(0, - 4)$

Step 1.3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- x - 4$ .

$y \leq - x - 4$

Step 2

Graph $- 3 x + 3 y \leq - 9$ .

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Step 2.1

Write in $y = m x + b$ form.

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Step 2.1.1

Solve for $y$ .

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Step 2.1.1.1

Add $3 x$ to both sides of the inequality.

$3 y \leq - 9 + 3 x$

Step 2.1.1.2

Divide each term in $3 y \leq - 9 + 3 x$ by $3$ and simplify.

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Step 2.1.1.2.1

Divide each term in $3 y \leq - 9 + 3 x$ by $3$ .

$\frac{3 y}{3} \leq \frac{- 9}{3} + \frac{3 x}{3}$

Step 2.1.1.2.2

Simplify the left side.

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Step 2.1.1.2.2.1

Cancel the common factor of $3$ .

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Step 2.1.1.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} \leq \frac{- 9}{3} + \frac{3 x}{3}$

Step 2.1.1.2.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{- 9}{3} + \frac{3 x}{3}$

Step 2.1.1.2.3

Simplify the right side.

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Step 2.1.1.2.3.1

Simplify each term.

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Step 2.1.1.2.3.1.1

Divide $- 9$ by $3$ .

$y \leq - 3 + \frac{3 x}{3}$

Step 2.1.1.2.3.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.2.3.1.2.1

Cancel the common factor.

$y \leq - 3 + \frac{3 x}{3}$

Step 2.1.1.2.3.1.2.2

Divide $x$ by $1$ .

$y \leq - 3 + x$

Step 2.1.2

Rearrange terms.

$y \leq x - 3$

Step 2.2

Use the slope-intercept form to find the slope and y-intercept.

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Step 2.2.1

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 2.2.2

Find the values of $m$ and $b$ using the form $y = m x + b$ .

$m = 1$

$b = - 3$

Step 2.2.3

The slope of the line is the value of $m$ , and the y-intercept is the value of $b$ .

Slope: $1$

y-intercept: $(0, - 3)$

Slope: $1$

y-intercept: $(0, - 3)$

Step 2.3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $x - 3$ .

$y \leq x - 3$

Step 3

Plot each graph on the same coordinate system.

$- y \geq x + 4$

$- 3 x + 3 y \leq - 9$

Step 4