Algebra Examples

$5 x - y < - 6$ $3 x - y \geq 4$

Step 1

Graph $5 x - y < - 6$ .

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Step 1.1

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

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Step 1.1.1

Solve for $y$ .

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Step 1.1.1.1

Subtract $5 x$ from both sides of the inequality.

$- y < - 6 - 5 x$

Step 1.1.1.2

Divide each term in $- y < - 6 - 5 x$ by $- 1$ and simplify.

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Step 1.1.1.2.1

Divide each term in $- y < - 6 - 5 x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} > \frac{- 6}{- 1} + \frac{- 5 x}{- 1}$

Step 1.1.1.2.2

Simplify the left side.

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Step 1.1.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} > \frac{- 6}{- 1} + \frac{- 5 x}{- 1}$

Step 1.1.1.2.2.2

Divide $y$ by $1$ .

$y > \frac{- 6}{- 1} + \frac{- 5 x}{- 1}$

Step 1.1.1.2.3

Simplify the right side.

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Step 1.1.1.2.3.1

Simplify each term.

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Step 1.1.1.2.3.1.1

Divide $- 6$ by $- 1$ .

$y > 6 + \frac{- 5 x}{- 1}$

Step 1.1.1.2.3.1.2

Move the negative one from the denominator of $\frac{- 5 x}{- 1}$ .

$y > 6 - 1 \cdot (- 5 x)$

Step 1.1.1.2.3.1.3

Rewrite $- 1 \cdot (- 5 x)$ as $- (- 5 x)$ .

$y > 6 - (- 5 x)$

Step 1.1.1.2.3.1.4

Multiply $- 5$ by $- 1$ .

$y > 6 + 5 x$

Step 1.1.2

Rearrange terms.

$y > 5 x + 6$

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 $y = m x + b$ .

$m = 5$

$b = 6$

Step 1.2.3

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

Slope: $5$

y-intercept: $(0, 6)$

Slope: $5$

y-intercept: $(0, 6)$

Step 1.3

Graph a dashed line, then shade the area above the boundary line since $y$ is greater than $5 x + 6$ .

$y > 5 x + 6$

Step 2

Graph $3 x - y \geq 4$ .

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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

Subtract $3 x$ from both sides of the inequality.

$- y \geq 4 - 3 x$

Step 2.1.1.2

Divide each term in $- y \geq 4 - 3 x$ by $- 1$ and simplify.

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Step 2.1.1.2.1

Divide each term in $- y \geq 4 - 3 x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

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

Step 2.1.1.2.2

Simplify the left side.

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Step 2.1.1.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.1.1.2.2.2

Divide $y$ by $1$ .

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

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 $4$ by $- 1$ .

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

Step 2.1.1.2.3.1.2

Move the negative one from the denominator of $\frac{- 3 x}{- 1}$ .

$y \leq - 4 - 1 \cdot (- 3 x)$

Step 2.1.1.2.3.1.3

Rewrite $- 1 \cdot (- 3 x)$ as $- (- 3 x)$ .

$y \leq - 4 - (- 3 x)$

Step 2.1.1.2.3.1.4

Multiply $- 3$ by $- 1$ .

$y \leq - 4 + 3 x$

Step 2.1.2

Rearrange terms.

$y \leq 3 x - 4$

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 $y = m x + b$ .

$m = 3$

$b = - 4$

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: $(0, - 4)$

Slope: $3$

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

Step 2.3

Graph a solid line, then shade the area below the boundary line since $y$ is less than $3 x - 4$ .

$y \leq 3 x - 4$

Step 3

Plot each graph on the same coordinate system.

$5 x - y < - 6$

$3 x - y \geq 4$

Step 4