Algebra Examples

$y \leq - \frac{2}{3} x + 3$ $y \leq \frac{3}{2} x$

Step 1

Graph $y \leq - \frac{2}{3} x + 3$ .

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Step 1.1

Find the slope and the y-intercept for the boundary line.

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Step 1.1.1

Rewrite in slope-intercept form.

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Step 1.1.1.1

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 1.1.1.2

Simplify the right side.

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Step 1.1.1.2.1

Simplify each term.

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Step 1.1.1.2.1.1

Combine $x$ and $\frac{2}{3}$ .

$y \leq - \frac{x \cdot 2}{3} + 3$

Step 1.1.1.2.1.2

Move $2$ to the left of $x$ .

$y \leq - \frac{2 x}{3} + 3$

Step 1.1.1.3

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

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Step 1.1.1.3.1

Reorder terms.

$y \leq - (\frac{2}{3} x) + 3$

Step 1.1.1.3.2

Remove parentheses.

$y \leq - \frac{2}{3} x + 3$

Step 1.1.2

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

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Step 1.1.2.1

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

$m = - \frac{2}{3}$

$b = 3$

Step 1.1.2.2

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

Slope: $- \frac{2}{3}$

y-intercept: $(0, 3)$

Slope: $- \frac{2}{3}$

y-intercept: $(0, 3)$

Slope: $- \frac{2}{3}$

y-intercept: $(0, 3)$

Step 1.2

Graph a solid line, then shade the area below the boundary line since $y$ is less than $- \frac{2}{3} x + 3$ .

$y \leq - \frac{2}{3} x + 3$

Step 2

Graph $y \leq \frac{3}{2} x$ .

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Step 2.1

Find the slope and the y-intercept for the boundary line.

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Step 2.1.1

Rewrite in slope-intercept form.

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Step 2.1.1.1

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 2.1.1.2

Simplify the right side.

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Step 2.1.1.2.1

Combine $\frac{3}{2}$ and $x$ .

$y \leq \frac{3 x}{2}$

Step 2.1.1.3

Reorder terms.

$y \leq \frac{3}{2} x$

Step 2.1.2

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

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Step 2.1.2.1

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

$m = \frac{3}{2}$

$b = 0$

Step 2.1.2.2

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

Slope: $\frac{3}{2}$

y-intercept: $(0, 0)$

Slope: $\frac{3}{2}$

y-intercept: $(0, 0)$

Slope: $\frac{3}{2}$

y-intercept: $(0, 0)$

Step 2.2

Graph a solid line, then shade the area below the boundary line since $y$ is less than $\frac{3}{2} x$ .

$y \leq \frac{3}{2} x$

Step 3

Plot each graph on the same coordinate system.

$y \leq - \frac{2}{3} x + 3$

$y \leq \frac{3}{2} x$

Step 4