Graphs

38 Graphs of Linear Inequalities

Learning Objectives

By the end of this section, you will be able to:

  • Verify solutions to an inequality in two variables
  • Recognize the relation between the solutions of an inequality and its graph
  • Graph linear inequalities

Before you get started, take this readiness quiz.

  1. Solve: 4x+3>23.
    If you missed this problem, review (Figure).
  2. Translate from algebra to English: x<5.
    If you missed this problem, review (Figure).
  3. Evaluate 3x-2y when x=1, y=-2.
    If you missed this problem, review (Figure).

Verify Solutions to an Inequality in Two Variables

We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit.

Linear Inequality

A linear inequality is an inequality that can be written in one of the following forms:

\begin{array}{cccccccccc}\hfill Ax+By>C\hfill & & & \hfill Ax+By\ge C\hfill & & & \hfill Ax+By<C\hfill & & & \hfill Ax+By\le C\hfill \end{array}

where A\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}B are not both zero.

Do you remember that an inequality with one variable had many solutions? The solution to the inequality x>3 is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See (Figure).

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity.

Similarly, inequalities in two variables have many solutions. Any ordered pair \left(x,y\right) that makes the inequality true when we substitute in the values is a solution of the inequality.

Solution of a Linear Inequality

An ordered pair \left(x,y\right) is a solution of a linear inequality if the inequality is true when we substitute the values of x and y.

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

\left(0,0\right)\left(1,6\right)\left(2,6\right)\left(-5,-15\right)\left(-8,12\right)

Solution

  1. \left(0,0\right) .
    . .
    Simplify. .
    So, \left(0,0\right) is not a solution to y>x+4.

  2. \left(1,6\right) .
    . .
    Simplify. .
    So, \left(1,6\right) is a solution to y>x+4.

  3. \left(2,6\right) .
    . .
    Simplify. .
    So, \left(2,6\right) is not a solution to y>x+4.

  4. \left(-5,-15\right) .
    . .
    Simplify. .
    So, \left(-5,-15\right) is not a solution to y>x+4.

  5. \left(-8,12\right) .
    . .
    Simplify. .
    So, \left(-8,12\right) is a solution to y>x+4.

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)

yes  yes  yes  yes  no

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)

yes  yes  no  no  yes

Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in (Figure) again. The point x=3 separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See (Figure).

The figure shows a number line extending from negative 5 to 5. A parenthesis is shown at positive 3 and an arrow extends form positive 3 to positive infinity. An arrow above the number line extends from 3 and points to the left. It is labeled “numbers less than 3.” An arrow above the number line extends from 3 and points to the right. It is labeled “numbers greater than 3.”

The solution to x>3 is the shaded part of the number line to the right of x=3.

Similarly, the line y=x+4 separates the plane into two regions. On one side of the line are points with y<x+4. On the other side of the line are the points with y>x+4. We call the line y=x+4 a boundary line.

Boundary Line

The line with equation Ax+By=C is the boundary line that separates the region where Ax+By>C from the region where Ax+By<C.

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not a is included in the solution:

The figure shows two number lines. The number line on the left is labeled x is less than a. The number line shows a parenthesis at a and an arrow that points to the left. The number line on the right is labeled x is less than or equal to a. The number line shows a bracket at a and an arrow that points to the left.

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to indicate whether or not it the line is included in the solution. This is summarized in (Figure)

Ax+By<C Ax+By\le C
Ax+By>C Ax+By\ge C
Boundary line is not included in solution. Boundary line is included in solution.
Boundary line is dashed. Boundary line is solid.

Now, let’s take a look at what we found in (Figure). We’ll start by graphing the line y=x+4, and then we’ll plot the five points we tested. See (Figure).

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15).

In (Figure) we found that some of the points were solutions to the inequality y>x+4 and some were not.

Which of the points we plotted are solutions to the inequality y>x+4? The points \left(1,6\right) and \left(-8,12\right) are solutions to the inequality y>x+4. Notice that they are both on the same side of the boundary line y=x+4.

The two points \left(0,0\right) and \left(-5,-15\right) are on the other side of the boundary line y=x+4, and they are not solutions to the inequality y>x+4. For those two points, y<x+4.

What about the point \left(2,6\right)? Because 6=2+4, the point is a solution to the equation y=x+4. So the point \left(2,6\right) is on the boundary line.

Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality y>x+4. The point \left(0,10\right) clearly looks to be to the left of the boundary line, doesn’t it? Is it a solution to the inequality?

\begin{array}{cccc}y>x+4\hfill & & & \\ 10\stackrel{?}{>}0+4\hfill & & & \\ 10>4\hfill & & & \text{So},\phantom{\rule{0.2em}{0ex}}\left(0,10\right)\phantom{\rule{0.2em}{0ex}}\text{is a solution to}\phantom{\rule{0.2em}{0ex}}y>x+4.\hfill \end{array}

Any point you choose on the left side of the boundary line is a solution to the inequality y>x+4. All points on the left are solutions.

Similarly, all points on the right side of the boundary line, the side with \left(0,0\right) and \left(-5,-15\right), are not solutions to y>x+4. See (Figure).

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15). To the upper left of the line is the inequality y is greater than x plus 4. To the right of the line is the inequality y is less than x plus 4.

The graph of the inequality y>x+4 is shown in (Figure) below. The line y=x+4 divides the plane into two regions. The shaded side shows the solutions to the inequality y>x+4.

The points on the boundary line, those where y=x+4, are not solutions to the inequality y>x+4, so the line itself is not part of the solution. We show that by making the line dashed, not solid.

The graph of the inequality y>x+4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as a dashed arrow extending from the bottom left toward the upper right. The coordinate plane to the upper left of the line is shaded.

The boundary line shown is y=2x-1. Write the inequality shown by the graph.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 x minus 1 is plotted as a solid arrow extending from the bottom left toward the upper right. The coordinate plane to the left of the line is shaded

Solution

The line y=2x-1 is the boundary line. On one side of the line are the points with y>2x-1 and on the other side of the line are the points with y<2x-1.

Let’s test the point \left(0,0\right) and see which inequality describes its side of the boundary line.

At \left(0,0\right), which inequality is true:

\begin{array}{ccccc}\hfill y>2x-1\hfill & & \hfill \text{or}\hfill & & \hfill y<2x-1?\hfill \\ \hfill y>2x-1\hfill & & & & \hfill y<2x-1\hfill \\ \hfill 0\stackrel{?}{>}2·0-1\hfill & & & & \hfill 0\stackrel{?}{<}2·0-1\hfill \\ \hfill 0>-1\phantom{\rule{0.2em}{0ex}}\text{True}\hfill & & & & \hfill 0<-1\phantom{\rule{0.2em}{0ex}}\text{False}\hfill \end{array}

Since, y>2x-1 is true, the side of the line with \left(0,0\right), is the solution. The shaded region shows the solution of the inequality y>2x-1.

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality y\ge 2x-1.

We could use any point as a test point, provided it is not on the line. Why did we choose \left(0,0\right)? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that y<2x-1.

Write the inequality shown by the graph with the boundary line y=-2x+3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x plus 3 is plotted as a solid arrow extending from the top left toward the bottom right. The coordinate plane to the right of the line is shaded.

y\ge -2x+3

Write the inequality shown by the graph with the boundary line y=\frac{1}{2}x-4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals one half x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the bottom right of the line is shaded.

y<\frac{1}{2}x-4

The boundary line shown is 2x+3y=6. Write the inequality shown by the graph.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus 3 y equals 6 is plotted as a dashed arrow extending from the top left toward the bottom right. The coordinate plane to the bottom of the line is shaded.

Solution

The line 2x+3y=6 is the boundary line. On one side of the line are the points with 2x+3y>6 and on the other side of the line are the points with 2x+3y<6.

Let’s test the point \left(0,0\right) and see which inequality describes its side of the boundary line.

At \left(0,0\right), which inequality is true:

\begin{array}{ccccccccccc}\hfill 2x+3y& >\hfill & 6\hfill & & & \hfill \text{or}\hfill & & & \hfill 2x+3y& <\hfill & 6?\hfill \\ \hfill 2x+3y& >\hfill & 6\hfill & & & & & & \hfill 2x+3y& <\hfill & 6\hfill \\ \hfill 2\left(0\right)+3\left(0\right)& \stackrel{?}{>}\hfill & 6\hfill & & & & & & \hfill 2\left(0\right)+3\left(0\right)& \stackrel{?}{<}\hfill & 6\hfill \\ \hfill 0& >\hfill & 6\phantom{\rule{0.2em}{0ex}}\text{False}\hfill & & & & & & \hfill 0& <\hfill & 6\phantom{\rule{0.2em}{0ex}}\text{True}\hfill \end{array}

So the side with \left(0,0\right) is the side where 2x+3y<6.

(You may want to pick a point on the other side of the boundary line and check that 2x+3y>6.)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The graph shows the solution to the inequality 2x+3y<6.

Write the inequality shown by the shaded region in the graph with the boundary line x-4y=8.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 4 y equals 8 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the top of the line is shaded.

x-4y\le 8

Write the inequality shown by the shaded region in the graph with the boundary line 3x-y=6.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the right of the line is shaded.

3x-y\le 6

Graph Linear Inequalities

Now, we’re ready to put all this together to graph linear inequalities.

How to Graph Linear Inequalities

Graph the linear inequality y\ge \frac{3}{4}x-2.

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Identify and graph the boundary line. If the inequality is less than or equal to or greater than or equal to, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. The text in the second cell reads: “Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y equals three-fourths x minus 2. The inequality sign is greater than or equal to, so we draw a solid line. The third cell contains the graph of the line three-fourths x minus 2 on a coordinate plane.In the second row of the table, the first cell says: “Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality? In the second cell, the instructions say: “We’ll test (0, 0). Is it a solution of the inequality?” The third cell asks: At (0, 0), is y greater than or equal to three-fourths x minus 2? Below that is the inequality 0 is greater than or equal to three-fourths 0 minus 2, with a question mark above the inequality symbol. Below that is the inequality 0 is greater than or equal to negative 2. Below that is: “So (0, 0) is a solution.In the third row of the table, the first cell says: “Step 3. Shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. In the second cell, the instructions say: The test point (0, 0) is a solution to y is greater than or equal to three-fourths x minus 2. So we shade in that side.” In the third cell is the graph of the line three-fourths x minus 2 on a coordinate plane with the region above the line shaded.

Graph the linear inequality y\ge \frac{5}{2}x-4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-halves x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.

Graph the linear inequality y<\frac{2}{3}x-5.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals two-thirds x minus 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.

The steps we take to graph a linear inequality are summarized here.

Graph a linear inequality.
  1. Identify and graph the boundary line.
    • If the inequality is \le \text{or}\ge, the boundary line is solid.
    • If the inequality is < or >, the boundary line is dashed.
  2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  3. Shade in one side of the boundary line.
    • If the test point is a solution, shade in the side that includes the point.
    • If the test point is not a solution, shade in the opposite side.

Graph the linear inequality x-2y<5.

Solution

First we graph the boundary line x-2y=5. The inequality is < so we draw a dashed line.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right.

Then we test a point. We’ll use \left(0,0\right) again because it is easy to evaluate and it is not on the boundary line.

Is \left(0,0\right) a solution of x-2y<5?

The figure shows the inequality 0 minus 2 times 0 in parentheses is less than 5, with a question mark above the inequality symbol. The next line shows 0 minus 0 is less than 5, with a question mark above the inequality symbol. The third line shows 0 is less than 5.

The point \left(0,0\right) is a solution of x-2y<5, so we shade in that side of the boundary line.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The point (0, 0) is plotted, but not labeled. The region above the line is shaded.

Graph the linear inequality 2x-3y\le 6.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 3 y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.

Graph the linear inequality 2x-y>3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 3 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.

What if the boundary line goes through the origin? Then we won’t be able to use \left(0,0\right) as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Graph the linear inequality y\le -4x.

Solution

First we graph the boundary line y=-4x. It is in slope–intercept form, with m=-4\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b=0. The inequality is \le so we draw a solid line.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line s y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right.

Now, we need a test point. We can see that the point \left(1,0\right) is not on the boundary line.

Is \left(1,0\right) a solution of y\le -4x?

The figure shows 0 is less than or equal to negative 4 times 1 in parentheses, with a question mark above the inequality symbol. The next line shows 0 is not less than or equal to negative 4.

The point \left(1,0\right) is not a solution to y\le -4x, so we shade in the opposite side of the boundary line. See (Figure).

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right. The point (1, 0) is plotted, but not labeled. The region to the left of the line is shaded.

Graph the linear inequality y>-3x.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a dashed arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.

Graph the linear inequality y\ge -2x.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x is plotted as a solid arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.

Some linear inequalities have only one variable. They may have an x but no y, or a y but no x. In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember?

\begin{array}{cccc}x=a\hfill & & & \text{vertical line}\hfill \\ y=b\hfill & & & \text{horizontal line}\hfill \end{array}

Graph the linear inequality y>3.

Solution

First we graph the boundary line y=3. It is a horizontal line. The inequality is > so we draw a dashed line.

We test the point \left(0,0\right).

\begin{array}{}\\ y>3\hfill \\ \\ 0\overline{)>}3\hfill \end{array}

\left(0,0\right) is not a solution to y>3.

So we shade the side that does not include (0, 0).

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.

Graph the linear inequality y<5.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 5 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.

Graph the linear inequality y\le -1.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 1 is plotted as a dashed arrow horizontally across the plane. The region below the line is shaded.

Key Concepts

  • To Graph a Linear Inequality
    1. Identify and graph the boundary line.
      If the inequality is \le \text{or}\ge, the boundary line is solid.
      If the inequality is < or >, the boundary line is dashed.
    2. Test a point that is not on the boundary line. Is it a solution of the inequality?
    3. Shade in one side of the boundary line.
      If the test point is a solution, shade in the side that includes the point.
      If the test point is not a solution, shade in the opposite side.

Section Exercises

Practice Makes Perfect

Verify Solutions to an Inequality in Two Variables

In the following exercises, 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-1:

\left(0,1\right)
\left(-4,-1\right)
\left(4,2\right)
\left(3,0\right)
\left(-2,-3\right)

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

\left(0,0\right)
\left(2,1\right)
\left(-1,-5\right)
\left(-6,-3\right)
\left(1,0\right)

yes  no  no  yes  no

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

\left(0,3\right)
\left(-3,-2\right)
\left(-2,0\right)
\left(0,0\right)
\left(-1,4\right)

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

\left(-3,0\right)
\left(1,6\right)
\left(-6,-2\right)
\left(0,1\right)
\left(5,-4\right)

yes  no  no  yes  yes

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

\left(5,1\right)
\left(-2,6\right)
\left(3,2\right)
\left(10,-5\right)
\left(0,0\right)

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

\left(1,1\right)
\left(4,-3\right)
\left(0,0\right)
\left(-8,12\right)
\left(3,0\right)

no  no  no  yes  yes

Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

Write the inequality shown by the graph with the boundary line y=3x-4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3x minus 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the right of the line is shaded.

Write the inequality shown by the graph with the boundary line y=2x-4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2x minus 4 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

y<2x-4

Write the inequality shown by the graph with the boundary line y=-\frac{1}{2}x+1.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-half x plus 1 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

Write the inequality shown by the graph with the boundary line y=-\frac{1}{3}x-2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

y\le -\frac{1}{3}x-2

Write the inequality shown by the shaded region in the graph with the boundary line x+y=5.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 5 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.

Write the inequality shown by the shaded region in the graph with the boundary line x+y=3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 3 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.

x+y\ge 3

Write the inequality shown by the shaded region in the graph with the boundary line 2x+y=-4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus y equals negative 4 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

Write the inequality shown by the shaded region in the graph with the boundary line x+2y=-2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 2 y equals negative 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

x+2y\ge -2

Write the inequality shown by the shaded region in the graph with the boundary line 3x-y=6.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

Write the inequality shown by the shaded region in the graph with the boundary line 2x-y=4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

2x-y<4

Write the inequality shown by the shaded region in the graph with the boundary line 2x-5y=10.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 5 y equals 10 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

Write the inequality shown by the shaded region in the graph with the boundary line 4x-3y=12.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x minus 3 y equals 12 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

4x-3y>12

Graph Linear Inequalities

In the following exercises, graph each linear inequality.

Graph the linear inequality y>\frac{2}{3}x-1.

Graph the linear inequality y<\frac{3}{5}x+2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals three-fifths x plus 2 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

Graph the linear inequality y\le -\frac{1}{2}x+4.

Graph the linear inequality y\ge -\frac{1}{3}x-2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

Graph the linear inequality x-y\le 3.

Graph the linear inequality x-y\ge -2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 2 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

Graph the linear inequality 4x+y>-4.

Graph the linear inequality x+5y<-5.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus 5 y equals negative 5 is plotted as a dashed line extending from the top left toward the bottom right. The region below the line is shaded.

Graph the linear inequality 3x+2y\ge -6.

Graph the linear inequality 4x+2y\ge -8.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x plus 2 y equals negative 8 is plotted as a solid line extending from the top left toward the bottom right. The region to the right of the line is shaded.

Graph the linear inequality y>4x.

Graph the linear inequality y>x.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x is plotted as a solid line extending from the bottom left toward the top right. The region above the line is shaded.

Graph the linear inequality y\le \text{−}x.

Graph the linear inequality y\le -3x.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a solid line extending from the top left toward the bottom right. The region to the left of the line is shaded.

Graph the linear inequality y\ge -2.

Graph the linear inequality y<-1.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 1 is plotted as a dashed horizontal line. The region below the line is shaded.

Graph the linear inequality y<4.

Graph the linear inequality y\ge 2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 2 is plotted as a solid horizontal line. The region above the line is shaded.

Graph the linear inequality x\le 5.

Graph the linear inequality x>-2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 2 is plotted as a dashed vertical line. The region to the right of the line is shaded.

Graph the linear inequality x>-3.

Graph the linear inequality x\le 4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals 4 is plotted as a solid vertical line. The region to the left of the line is shaded.

Graph the linear inequality x-y<4.

Graph the linear inequality x-y<-3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus y equals negative 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.

Graph the linear inequality y\ge \frac{3}{2}x.

Graph the linear inequality y\le \frac{5}{4}x.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-fourths x is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

Graph the linear inequality y>-2x+1.

Graph the linear inequality y<-3x-4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x minus 4 is plotted as a dashed line extending from the top left toward the bottom right. The region to the left of the line is shaded.

Graph the linear inequality x\le -1.

Graph the linear inequality x\ge 0.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x equals negative 0 is plotted as a solid vertical line along the y-axis. The region to the right of the line is shaded.

Everyday Math

Money. Gerry wants to have a maximum of ?100 cash at the ticket booth when his church carnival opens. He will have ?1 bills and ?5 bills. If x is the number of ?1 bills and y is the number of ?5 bills, the inequality x+5y\le 100 models the situation.

  1. Graph the inequality.
  2. List three solutions to the inequality x+5y\le 100 where both x and y are integers.

Shopping. Tula has ?20 to spend at the used book sale. Hardcover books cost ?2 each and paperback books cost ?0.50 each. If x is the number of hardcover books Tula can buy and y is the number of paperback books she can buy, the inequality 2x+\frac{1}{2}y\le 20 models the situation.

  1. Graph the inequality.
  2. List three solutions to the inequality 2x+\frac{1}{2}y\le 20 where both x and y are whole numbers.

  1. The graph shows the x y-coordinate plane. The x- axis runs from 0 to 20 and the y-axis runs from 0 to 30. The line 2 x plus one-half y equals 20 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.
  2. Answers will vary.

Writing Exercises

Lester thinks that the solution of any inequality with a > sign is the region above the line and the solution of any inequality with a < sign is the region below the line. Is Lester correct? Explain why or why not.

Explain why in some graphs of linear inequalities the boundary line is solid but in other graphs it is dashed.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right: “I can…,” “confidently,” “with some help,” and “no-I don’t get it!” The first column below “I can…” reads “verify solutions to an inequality in two variables,”, “recognize the relation between the solutions of an inequality and its graph,” and “graph linear inequalities.” The rest of the cells are blank.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Chapter 4 Review Exercises

Rectangular Coordinate System

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system.


\left(-1,-5\right)
\left(-3,4\right)
\left(2,-3\right)
\left(1,\frac{5}{2}\right)


\left(4,3\right)
\left(-4,3\right)
\left(-4,-3\right)
\left(4,-3\right)

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (4, 3) is plotted and labeled "a". The point (negative 4, 3) is plotted and labeled "b". The point (negative 4, negative 3) is plotted and labeled "c". The point (4, negative 3) is plotted and labeled “d”.


\left(-2,0\right)
\left(0,-4\right)
\left(0,5\right)
\left(3,0\right)


\left(2,\frac{3}{2}\right)
\left(3,\frac{4}{3}\right)
\left(\frac{1}{3},-4\right)
\left(\frac{1}{2},-5\right)

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (2, three halves) is plotted and labeled "a". The point (3, four thirds) is plotted and labeled "b". The point (one third, negative 4) is plotted and labeled "c". The point (one-half, negative 5) is plotted and labeled “d”.

Identify Points on a Graph

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (5, 3) is plotted and labeled "a". The point (2, negative 1) is plotted and labeled "b". The point (negative 3, negative 2) is plotted and labeled "c". The point (negative 1, 4) is plotted and labeled “d”.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 6 to 6. The point (2, 0) is plotted and labeled "a". The point (0, negative 5) is plotted and labeled "b". The point (negative 4, 0) is plotted and labeled "c". The point (0, 3) is plotted and labeled “d”.

\left(2,0\right)\left(0,-5\right)\left(-4.0\right)\left(0,3\right)

Verify Solutions to an Equation in Two Variables

In the following exercises, which ordered pairs are solutions to the given equations?

5x+y=10

\left(5,1\right)
\left(2,0\right)
\left(4,-10\right)

y=6x-2

\left(1,4\right)
\left(\frac{1}{3},0\right)
\left(6,-2\right)

a, b

Complete a Table of Solutions to a Linear Equation in Two Variables

In the following exercises, complete the table to find solutions to each linear equation.

y=4x-1

x y \left(x,y\right)
0
1
-2

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

x y \left(x,y\right)
0
4
-2
x y \left(x,y\right)
0 3 \left(0,3\right)
4 1 (4, 1)
-2 4 \left(-2,4\right)

x+2y=5

x y \left(x,y\right)
0
1
-1

3x+2y=6

x y \left(x,y\right)
0
0
-2
x y \left(x,y\right)
0 -3 \left(0,-3\right)
2 0 \left(2,0\right)
-2 -6 \left(-2,-6\right)

Find Solutions to a Linear Equation in Two Variables

In the following exercises, find three solutions to each linear equation.

x+y=3

x+y=-4

Answers will vary.

y=3x+1

y=\text{−}x-1

Answers will vary.

Graphing Linear Equations

Recognize the Relation Between the Solutions of an Equation and its Graph

In the following exercises, for each ordered pair, decide:

  1. Is the ordered pair a solution to the equation?
  2. Is the point on the line?

y=\text{−}x+4

\left(0,4\right)\left(-1,3\right)

\left(2,2\right)\left(-2,6\right)

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 4 is plotted as an arrow extending from the top left toward the bottom right.

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

\left(0,-1\right) (3, 1)

\left(-3,-3\right) (6, 4)

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 1 is plotted as an arrow extending from the bottom left toward the top right.

yes; yes  yes; no

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

y=4x-3

y=-3x

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 3 x is plotted as an arrow extending from the top left toward the bottom right.

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

x-y=6

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus y equals 6 is plotted as an arrow extending from the bottom left toward the top right.

2x+y=7

3x-2y=6

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 3 x minus 2 y equals 6 is plotted as an arrow extending from the bottom left toward the top right.

Graph Vertical and Horizontal lines

In the following exercises, graph each equation.

y=-2

x=3

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x equals 3 is plotted as a vertical line.

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

y=-2x and y=-2

y=\frac{4}{3}x and y=\frac{4}{3}

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals four-thirds x is plotted as an arrow extending from the bottom left toward the top right. The line y equals four-thirds is plotted as a horizontal line.

Graphing with Intercepts

Identify the x– and y-Intercepts on a Graph

In the following exercises, find the x– and y-intercepts.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (3, 0) and (0, 3) is plotted.

\left(3,0\right),\left(0,3\right)

Find the x– and y-Intercepts from an Equation of a Line

In the following exercises, find the intercepts of each equation.

x+y=5

x-y=-1

\left(-1,0\right),\left(0,1\right)

x+2y=6

2x+3y=12

\left(6,0\right),\left(0,4\right)

y=\frac{3}{4}x-12

y=3x

\left(0,0\right)

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

\text{−}x+3y=3

x+y=-2

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 2 is plotted as an arrow extending from the top left toward the bottom right.

x-y=4

2x-y=5

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 2 x minus y equals 5 is plotted as an arrow extending from the bottom left toward the top right.

2x-4y=8

y=2x

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 x is plotted as an arrow extending from the bottom left toward the top right.

Slope of a Line

Use Geoboards to Model Slope

In the following exercises, find the slope modeled on each geoboard.

The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.
The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.

\frac{4}{3}

The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.
The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.

-\frac{2}{3}

In the following exercises, model each slope. Draw a picture to show your results.

\frac{1}{3}

\frac{3}{2}

The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 3 row 2.

-\frac{2}{3}

-\frac{1}{2}

The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 2 row 2 and the point in column 3 row 3.

Use m=\frac{\text{rise}}{\text{run}} to find the Slope of a Line from its Graph

In the following exercises, find the slope of each line shown.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 1, 3), (0, 0), and (1, negative 3) is plotted.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 4, 0) and (0, 4) is plotted.

1

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 6) and (5, 2) is plotted.

-\frac{1}{2}

Find the Slope of Horizontal and Vertical Lines

In the following exercises, find the slope of each line.

y=2

x=5

undefined

x=-3

y=-1

0

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

\left(-1,-1\right),\left(0,5\right)

\left(3,5\right),\left(4,-1\right)

-6

\left(-5,-2\right),\left(3,2\right)

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

\frac{5}{2}

Graph a Line Given a Point and the Slope

In the following exercises, graph each line with the given point and slope.

\left(2,-2\right); m=\frac{5}{2}

\left(-3,4\right); m=-\frac{1}{3}

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 3, 4) and (0, 3) is plotted.

x-intercept -4; m=3

y-intercept 1; m=-\frac{3}{4}

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (0, 1) and (4, negative 2) is plotted.

Solve Slope Applications

In the following exercises, solve these slope applications.

The roof pictured below has a rise of 10 feet and a run of 15 feet. What is its slope?

The figure shows a person on a ladder using a hammer on the roof of a building.

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

\frac{1}{10}

Intercept Form of an Equation of a Line

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

In the following exercises, use the graph to find the slope and y-intercept of each line. Compare the values to the equation y=mx+b.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 4 x minus 1 is plotted from the lower left to the top right.

y=4x-1

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x plus 4 is plotted from the top left to the bottom right.

y=-\frac{2}{3}x+4

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

Identify the Slope and y-Intercept from an Equation of a Line

In the following exercises, identify the slope and y-intercept of each line.

y=-4x+9

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

\frac{5}{3};\left(0,-6\right)

5x+y=10

4x-5y=8

\frac{4}{5};\left(0,-\frac{8}{5}\right)

Graph a Line Using Its Slope and Intercept

In the following exercises, graph the line of each equation using its slope and y-intercept.

y=2x+3

y=\text{−}x-1

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x minus 1 is plotted from the top left to the bottom right.

y=-\frac{2}{5}x+3

4x-3y=12

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 4 x minus 3 y equals 12 is plotted from the bottom left to the top right.

In the following exercises, determine the most convenient method to graph each line.

x=5

y=-3

horizontal line

2x+y=5

x-y=2

intercepts

y=x+2

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

plotting points

Graph and Interpret Applications of Slope–Intercept

Katherine is a private chef. The equation C=6.5m+42 models the relation between her weekly cost, C, in dollars and the number of meals, m, that she serves.

  1. Find Katherine’s cost for a week when she serves no meals.
  2. Find the cost for a week when she serves 14 meals.
  3. Interpret the slope and C-intercept of the equation.
  4. Graph the equation.

Marjorie teaches piano. The equation P=35h-250 models the relation between her weekly profit, P, in dollars and the number of student lessons, s, that she teaches.

  1. Find Marjorie’s profit for a week when she teaches no student lessons.
  2. Find the profit for a week when she teaches 20 student lessons.
  3. Interpret the slope and P–intercept of the equation.
  4. Graph the equation.

−?250  ?450  The slope, 35, means that Marjorie’s weekly profit, P, increases by ?35 for each additional student lesson she teaches. The P–intercept means that when the number of lessons is 0, Marjorie loses ?250. 

The graph shows the x y-coordinate plane where h is plotted along the x-axis and P is potted along the y-axis. The x-axis runs from 0 to 24. The y-axis runs from negative 300 to 500. The line P equals 35 h minus 250 is plotted from the bottom left to the top right.

Use Slopes to Identify Parallel Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.

4x-3y=-1;\phantom{\rule{0.2em}{0ex}}y=\frac{4}{3}x-3

2x-y=8;\phantom{\rule{0.2em}{0ex}}x-2y=4

not parallel

Use Slopes to Identify Perpendicular Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.

y=5x-1;10x+2y=0

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

perpendicular

Find the Equation of a Line

Find an Equation of the Line Given the Slope and y-Intercept

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

slope \frac{1}{3} and y\text{-intercept} \left(0,-6\right)

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

y=-5x-3

slope 0 and y\text{-intercept} \left(0,4\right)

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

y=-2x

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 x plus 1 is plotted from the bottom left to the top right.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 3 x plus 5 is plotted from the top left to the bottom right.

y=-3x+5

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals three-fourths x minus 2 is plotted from the bottom left to the top right.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 4 is plotted as a horizontal line.

y=-4

Find an Equation of the Line Given the Slope and a Point

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

m=-\frac{1}{4}, point \left(-8,3\right)

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

y=\frac{3}{5}x

Horizontal line containing \left(-2,7\right)

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

y=-2x-5

Find an Equation of the Line Given Two Points

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.

\left(2,10\right) and \left(-2,-2\right)

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

y=\frac{1}{2}x-\frac{5}{2}

\left(3,8\right) and \left(3,-4\right).

\left(5,2\right) and \left(-1,2\right)

y=2

Find an Equation of a Line Parallel to a Given Line

In the following exercises, 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=-3x+6, point \left(1,-5\right)

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

y=-\frac{2}{5}x+8

line x=4, point \left(-2,-1\right)

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

y=3

Find an Equation of a Line Perpendicular to a Given Line

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

line y=-\frac{4}{5}x+2, point \left(8,9\right)

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

y=-\frac{3}{2}x-6

line y=3, point \left(-1,-3\right)

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

y=1

Graph Linear Inequalities

Verify Solutions to an Inequality in Two Variables

In the following exercises, 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:

\left(0,1\right)
\left(-2,-4\right)
\left(5,2\right)
\left(3,-1\right)
\left(-1,-5\right)

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

\left(6,1\right)
\left(-3,6\right)
\left(3,2\right)
\left(-5,10\right)
\left(0,0\right)

yes  no  yes  yes  no

Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

Write the inequality shown by the graph with the boundary line y=\text{−}x+2.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x plus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

Write the inequality shown by the graph with the boundary line y=\frac{2}{3}x-3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals two-thirds x minus 3 is plotted as a dashed line extending from the bottom left toward the top right. The region above the line is shaded.

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

Write the inequality shown by the shaded region in the graph with the boundary line x+y=-4.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x plus y equals negative 4 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.

Write the inequality shown by the shaded region in the graph with the boundary line x-2y=6.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus 2 y equals 6 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

x-2y\ge 6

Graph Linear Inequalities

In the following exercises, graph each linear inequality.

Graph the linear inequality y>\frac{2}{5}x-4.

Graph the linear inequality y\le -\frac{1}{4}x+3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative one-fourth x plus 3 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

Graph the linear inequality x-y\le 5.

Graph the linear inequality 3x+2y>10.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line 3 x plus 2 y equals 10 is plotted as a dashed line extending from the top left toward the bottom right. The region above the line is shaded.

Graph the linear inequality y\le -3x.

Graph the linear inequality y<6.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 6 is plotted as a dashed, horizontal line. The region below the line is shaded.

Practice Test

Plot each point in a rectangular coordinate system.

\left(2,5\right)
\left(-1,-3\right)
\left(0,2\right)
\left(-4,\frac{3}{2}\right)
\left(5,0\right)

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

\left(3,3\right)
\left(2,0\right)
\left(4,-6\right)

yes  yes  no

Find three solutions to the linear equation y=-2x-4.

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

\left(3,0\right),\left(0,-4\right)

Find the slope of each line shown.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A line passing through the points (negative 5, 2) and (0, negative 1) is plotted from the top left toward the bottom right.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A vertical line passing through the point (2, 0) is plotted.

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The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. A horizontal line passing through the point (0, 5) is plotted.

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

1

Graph the line with slope \frac{1}{2} containing the point \left(-3,-4\right).

Graph the line for each of the following equations.

y=\frac{5}{3}x-1

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals five-thirds x minus 1 is plotted. The line passes through the points (0, negative 1) and (three-fifths, 0).

y=\text{−}x

x-y=2

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line x minus y equals 2 is plotted. The line passes through the points (0, negative 2) and (2, 0).

4x+2y=-8

y=2

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals 2 is plotted as a horizontal line passing through the point (0, 2).

x=-3

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

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

y=-\frac{3}{4}x-2

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

containing \left(10,1\right) and \left(6,-1\right)

y=\frac{1}{2}x-4

parallel to the line y=-\frac{2}{3}x-1, containing the point \left(-3,8\right)

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

y=-\frac{4}{5}x-5

Write the inequality shown by the graph with the boundary line y=\text{−}x-3.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative x minus 3 is plotted. The solid line passes through the points (negative 3, 0) and (0, negative 3).

Graph each linear inequality.

y>\frac{3}{2}x+5

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals three-halves x plus 5 is plotted. The dashed line passes through the points (0, 5) and (2, 8).

x-y\ge -4

y\le -5x

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 7 to 7. The line y equals negative 5 x is plotted. The solid line passes through the points (0, 0) and (1, negative 5).

y<3

Glossary

boundary line
The line with equation Ax+By=C that separates the region where Ax+By>C from the region where Ax+By<C.
linear inequality
An inequality that can be written in one of the following forms:

\begin{array}{cccccccccc}\hfill Ax+By>C\hfill & & & \hfill Ax+By\ge C\hfill & & & \hfill Ax+By<C\hfill & & & \hfill Ax+By\le C\hfill \end{array}


where A\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}B are not both zero.

solution of a linear inequality
An ordered pair \left(x,y\right) is a solution to a linear inequality the inequality is true when we substitute the values of x and y.

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Elementary Algebra by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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