4.7 Graphs of Linear Inequalities

Before you get started, take this readiness quiz.

Solve: $4 x + 3 > 23$.
If you missed this problem, review the reference.

Translate from algebra to English: $x < 5$.
If you missed this problem, review the reference.

Evaluate $3 x - 2 y$ when $x = 1$, $y = −2$.
If you missed this problem, review the reference.

Determine whether each ordered pair is a solution to the inequality $y > x - 3$:

ⓐ $\left(0 , 0\right)$ ⓑ $\left(4 , 9\right)$ ⓒ $\left(−2 , 1\right)$ ⓓ $\left(−5 , −3\right)$ ⓔ $\left(5 , 1\right)$

Determine whether each ordered pair is a solution to the inequality $y < x + 1$:

ⓐ $\left(0 , 0\right)$ ⓑ $\left(8 , 6\right)$ ⓒ $\left(−2 , −1\right)$ ⓓ $\left(3 , 4\right)$ ⓔ $\left(−1 , −4\right)$

Determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality $y > x - 3$:

1. ⓐ $\left(0 , 0\right)$
2. ⓑ $\left(2 , 1\right)$
3. ⓒ $\left(−1 , −5\right)$
4. ⓓ $\left(−6 , −3\right)$
5. ⓔ $\left(1 , 0\right)$

Determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality $y < x + 5$:

1. ⓐ $\left(−3 , 0\right)$
2. ⓑ $\left(1 , 6\right)$
3. ⓒ $\left(−6 , −2\right)$
4. ⓓ $\left(0 , 1\right)$
5. ⓔ $\left(5 , −4\right)$

Determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality $x + y > 2$:

1. ⓐ $\left(1 , 1\right)$
2. ⓑ $\left(4 , −3\right)$
3. ⓒ $\left(0 , 0\right)$
4. ⓓ $\left(−8 , 12\right)$
5. ⓔ $\left(3 , 0\right)$

Which ordered pairs are solutions to the given equations?

$y = 6 x - 2$

1. ⓐ $\left(1 , 4\right)$
2. ⓑ $\left(\frac{1}{3} , 0\right)$
3. ⓒ $\left(6 , −2\right)$

Complete the table to find solutions to each linear equation.

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

Complete the table to find solutions to each linear equation.

$3 x + 2 y = 6$

Find three solutions to each linear equation.

$x + y = −4$

Find three solutions to each linear equation.

$y = − x - 1$

Find the intercepts of each equation.

$x - y = −1$

Find the intercepts of each equation.

$2 x + 3 y = 12$

Find the intercepts of each equation.

$y = 3 x$

Find the slope of each line.

$x = 5$

Find the slope of each line.

$y = −1$

Use the slope formula to find the slope of the line between each pair of points.

$\left(3 , 5\right) , \left(4 , −1\right)$

Use the slope formula to find the slope of the line between each pair of points.

$\left(2 , 1\right) , \left(4 , 6\right)$

Solve these slope applications.

A mountain road rises 50 feet for a 500-foot run. What is its slope?

Identify the slope and y-intercept of each line.

$y = \frac{5}{3} x - 6$

Identify the slope and y-intercept of each line.

$4 x - 5 y = 8$

Determine the most convenient method to graph each line.

$y = −3$

Determine the most convenient method to graph each line.

$x - y = 2$

Determine the most convenient method to graph each line.

$y = \frac{3}{4} x - 1$

Use slopes and y-intercepts to determine if the lines are parallel.

$2 x - y = 8 ; x - 2 y = 4$

Use slopes and y-intercepts to determine if the lines are perpendicular.

$3 x - 2 y = 5 ; 2 x + 3 y = 6$

Find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

slope $−5$ and $y -\text{intercept}$ $\left(0 , −3\right)$

Find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

slope $−2$ and $y -\text{intercept}$ $\left(0 , 0\right)$

Find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

$m = \frac{3}{5}$, point $\left(10 , 6\right)$

Find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

$m = −2$, point $\left(−1 , −3\right)$

Find the equation of a line containing the given points. Write the equation in slope–intercept form.

$\left(7 , 1\right)$ and $\left(5 , 0\right)$

Find the equation of a line containing the given points. Write the equation in slope–intercept form.

$\left(5 , 2\right)$ and $\left(−1 , 2\right)$

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

line $2 x + 5 y = −10$, point $\left(10 , 4\right)$

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

line $y = −5$, point $\left(−4 , 3\right)$

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

line $2 x - 3 y = 9$, point $\left(−4 , 0\right)$

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

line $x = −5$ point $\left(2 , 1\right)$

Determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality $x + y > 4$:

1. ⓐ $\left(6 , 1\right)$
2. ⓑ $\left(−3 , 6\right)$
3. ⓒ $\left(3 , 2\right)$
4. ⓓ $\left(−5 , 10\right)$
5. ⓔ $\left(0 , 0\right)$

Which of the given ordered pairs are solutions to the equation $3 x - y = 6$?

1. ⓐ $\left(3 , 3\right)$
2. ⓑ $\left(2 , 0\right)$
3. ⓒ $\left(4 , −6\right)$

Find the x- and y-intercepts of the equation $4 x - 3 y = 12$.

Find the slope of each line shown.

Find the slope of the line between the points $\left(5 , 2\right)$ and $\left(−1 , −4\right)$.

Find the equation of each line. Write the equation in slope–intercept form.

slope $- \frac{3}{4}$ and y-intercept $\left(0 , −2\right)$

Find the equation of each line. Write the equation in slope–intercept form.

containing $\left(10 , 1\right)$ and $\left(6 , −1\right)$

Find the equation of each line. Write the equation in slope–intercept form.

perpendicular to the line $y = \frac{5}{4} x + 2$, containing the point $\left(−10 , 3\right)$