Algebra Examples

$y < 3 x + 2$ , $2 y \leq 6 x - 4$

Step 1

Graph $y < 3 x + 2$ .

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Step 1.1

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

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Step 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 1.1.2

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

$m = 3$

$b = 2$

Step 1.1.3

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

Slope: $3$

y-intercept: $(0, 2)$

Slope: $3$

y-intercept: $(0, 2)$

Step 1.2

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $3 x + 2$ .

$y < 3 x + 2$

Step 2

Graph $2 y \leq 6 x - 4$ .

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Step 2.1

Divide each term in $2 y \leq 6 x - 4$ by $2$ and simplify.

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Step 2.1.1

Divide each term in $2 y \leq 6 x - 4$ by $2$ .

$\frac{2 y}{2} \leq \frac{6 x}{2} + \frac{- 4}{2}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $2$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{2 y}{2} \leq \frac{6 x}{2} + \frac{- 4}{2}$

Step 2.1.2.1.2

Divide $y$ by $1$ .

$y \leq \frac{6 x}{2} + \frac{- 4}{2}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Cancel the common factor of $6$ and $2$ .

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Step 2.1.3.1.1.1

Factor $2$ out of $6 x$ .

$y \leq \frac{2 (3 x)}{2} + \frac{- 4}{2}$

Step 2.1.3.1.1.2

Cancel the common factors.

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Step 2.1.3.1.1.2.1

Factor $2$ out of $2$ .

$y \leq \frac{2 (3 x)}{2 (1)} + \frac{- 4}{2}$

Step 2.1.3.1.1.2.2

Cancel the common factor.

$y \leq \frac{2 (3 x)}{2 \cdot 1} + \frac{- 4}{2}$

Step 2.1.3.1.1.2.3

Rewrite the expression.

$y \leq \frac{3 x}{1} + \frac{- 4}{2}$

Step 2.1.3.1.1.2.4

Divide $3 x$ by $1$ .

$y \leq 3 x + \frac{- 4}{2}$

Step 2.1.3.1.2

Divide $- 4$ by $2$ .

$y \leq 3 x - 2$

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 $y = m x + b$ .

$m = 3$

$b = - 2$

Step 2.2.3

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

Slope: $3$

y-intercept: $(0, - 2)$

Slope: $3$

y-intercept: $(0, - 2)$

Step 2.3

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

$y \leq 3 x - 2$

Step 3

Plot each graph on the same coordinate system.

$y < 3 x + 2$

$2 y \leq 6 x - 4$

Step 4