Graph 8x-y>9 1/2x+6>=y y+1>-1/3(x-9)

$8 x - y > 9$ $\frac{1}{2} x + 6 \geq y$ $y + 1 > \frac{- 1}{3} (x - 9)$

Step 1

Graph $8 x - y > 9$ .

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

Solve for $y$ .

Subtract $8 x$ from both sides of the inequality.

$- y > 9 - 8 x$

Divide each term in $- y > 9 - 8 x$ by $- 1$ and simplify.

Divide each term in $- y > 9 - 8 x$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} < \frac{9}{- 1} + \frac{- 8 x}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} < \frac{9}{- 1} + \frac{- 8 x}{- 1}$

Divide $y$ by $1$ .

$y < \frac{9}{- 1} + \frac{- 8 x}{- 1}$

Simplify the right side.

Simplify each term.

Divide $9$ by $- 1$ .

$y < - 9 + \frac{- 8 x}{- 1}$

Move the negative one from the denominator of $\frac{- 8 x}{- 1}$ .

$y < - 9 - 1 \cdot (- 8 x)$

Rewrite $- 1 \cdot (- 8 x)$ as $- (- 8 x)$ .

$y < - 9 - (- 8 x)$

Multiply $- 8$ by $- 1$ .

$y < - 9 + 8 x$

Rearrange terms.

$y < 8 x - 9$

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

The slope-intercept form is $y = m x + b$ , where $m$ is the slope and $b$ is the y-intercept.

$y = m x + b$

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

$m = 8$

$b = - 9$

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

Slope: $8$

y-intercept: $(0, - 9)$

Slope: $8$

y-intercept: $(0, - 9)$

Graph a dashed line, then shade the area below the boundary line since $y$ is less than $8 x - 9$ .

$y < 8 x - 9$

Step 2

Graph $\frac{1}{2} x + 6 \geq y$ .

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

Solve for $y$ .

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

$y \leq \frac{1}{2} x + 6$

Combine $\frac{1}{2}$ and $x$ .

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

Reorder terms.

$y \leq \frac{1}{2} x + 6$

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

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

$m = \frac{1}{2}$

$b = 6$

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

Slope: $\frac{1}{2}$

y-intercept: $(0, 6)$

Slope: $\frac{1}{2}$

y-intercept: $(0, 6)$

Graph a solid line, then shade the area below the boundary line since $y$ is less than $\frac{1}{2} x + 6$ .

$y \leq \frac{1}{2} x + 6$

Step 3

Graph $y + 1 > \frac{- 1}{3} \cdot (x - 9)$ .

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

Solve for $y$ .

Simplify $\frac{- 1}{3} \cdot (x - 9)$ .

Rewrite.

$y + 1 > 0 + 0 + \frac{- 1}{3} \cdot (x - 9)$

Move the negative in front of the fraction.

$y + 1 > - \frac{1}{3} \cdot (x - 9)$

Apply the distributive property.

$y + 1 > - \frac{1}{3} x - \frac{1}{3} \cdot - 9$

Combine $x$ and $\frac{1}{3}$ .

$y + 1 > - \frac{x}{3} - \frac{1}{3} \cdot - 9$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$y + 1 > - \frac{x}{3} + \frac{- 1}{3} \cdot - 9$

Factor $3$ out of $- 9$ .

$y + 1 > - \frac{x}{3} + \frac{- 1}{3} \cdot (3 (- 3))$

Cancel the common factor.

$y + 1 > - \frac{x}{3} + \frac{- 1}{3} \cdot (3 \cdot - 3)$

Rewrite the expression.

$y + 1 > - \frac{x}{3} - 1 \cdot - 3$

Multiply $- 1$ by $- 3$ .

$y + 1 > - \frac{x}{3} + 3$

Move all terms not containing $y$ to the right side of the inequality.

Subtract $1$ from both sides of the inequality.

$y > - \frac{x}{3} + 3 - 1$

Subtract $1$ from $3$ .

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

Reorder terms.

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

Remove parentheses.

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

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

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

$m = - \frac{1}{3}$

$b = 2$

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

Slope: $- \frac{1}{3}$

y-intercept: $(0, 2)$

Slope: $- \frac{1}{3}$

y-intercept: $(0, 2)$

Graph a dashed line, then shade the area above the boundary line since $y$ is greater than $- \frac{1}{3} x + 2$ .

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

Step 4

Plot each graph on the same coordinate system.

$8 x - y > 9$

$\frac{1}{2} x + 6 \geq y$

$y + 1 > \frac{- 1}{3} \cdot (x - 9)$

Step 5

Graph 8x-y>9 1/2x+6>=y y+1>-1/3(x-9)

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