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Linear Inequalities in Two Variables

Graph inequalities like 2x + 3 < y on the coordinate plane

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Graphing Linear Inequalities
License: CC BY-NC 3.0

You are invited to a party and asked to bring chips and dips. Chips cost $2.50 a bag and a soda costs $1.50. If you only have $20.00 to spend, how many of each can you buy?

In this concept, you will learn to graph linear inequalities.

Graphing Linear Inequalities

The solutions to an inequality can be listed as ordered pairs. However, listing the ordered pairs would take a long time. Another way to represent the solution to an inequality is by graphing it on a number line. If the symbol shown in the inequality is \begin{align*}>\end{align*}> or < the endpoint is displayed using an open circle (an outline of a circle) since that value is not included in the solution. When the symbol shown in the inequality is \begin{align*}\ge\end{align*} or \begin{align*}\le\end{align*} the endpoint is displayed using a closed circle since that value is included in the solution.

Let’s look at an example.

Show the solution set for the given inequality on a number line,


First, draw a number line that shows numbers that are less than 2. Since 2 is the endpoint, 2 must be shown on the number line. It is a good idea to show at least one number to the right of the endpoint to create a clearer picture of the solution.

Next, mark the endpoint of 2 using an open circle.

Then, draw a line to the left with an arrow on the end to indicate the direction of the values in the solution set. The following diagram is the number line graph that models the given inequality:


License: CC BY-NC 3.0

This is only one method of visually displaying the solution set of an inequality and is used most often when the inequality consists of a single variable.

A linear inequality that has two variables can be plotted on a Cartesian grid the same way a linear equation can be plotted. The inequality can be written in slope-intercept form the same way an equation can be expressed in this form. The plotted line modeling a linear inequality serves as the boundary line between the solution set and the rest of the Cartesian grid. If the inequality displays the symbol \begin{align*}>\end{align*}> or < the boundary line will be a broken or dashed line. However, if the inequality displays the symbol \begin{align*}\ge\end{align*}  or \begin{align*}\le\end{align*}  the boundary line will be a solid line.

To determine which side of the boundary line contains the solution set, an ordered pair will be selected from one side of the line and its coordinates will be substituted into the given inequality. Do not use a point that the line passes through. If the ordered pair makes the inequality true then the side from which the point was selected will be shaded to show where the solution set is located on the graph. If the ordered pair does not make the inequality true, then the other side of the boundary line will be shaded.

Let’s look at an example.

Graph the solution set of the following inequality. 


Notice the inequality is written in slope-intercept form \begin{align*}y=mx+b\end{align*}y=mx+b with an inequality symbol replacing the equal sign.

First, plot the ‘\begin{align*}b\end{align*}b’ value, which is the \begin{align*}y\end{align*}y-intercept, and from here move horizontally one to the right and up one which is the value of the slope\begin{align*}m\end{align*}m’.

Next, join the two points using a broken line since the inequality symbol \begin{align*}>\end{align*}> is displayed.

Then, choose an ordered pair from one side of the boundary line to determine if it makes the inequality true.

Test \begin{align*}(1,1)\end{align*}(1,1)

\begin{align*} \begin{array}{rcl} y &>& x+3 \\ 1 &>& 1+3 \\ 1 &>& 4 \end{array} \end{align*}y11>>>x+31+34

The number 1 is not greater than 4. The test point \begin{align*}(1,1)\end{align*}(1,1) did not make the inequality true. Therefore, the other side of the boundary line must be shaded to indicate the location of the solution set.

License: CC BY-NC 3.0

An inequality that has two variables can be plotted in this way. Remember to display the boundary line with the correct style of line (broken or solid) as indicated by the inequality symbol. Then, shade the correct side of the boundary line according to the result of the test point.


Example 1

Earlier, you were given a problem about your invitation to a party. You have to figure out how many bags of chips and how many bottles of soda you can buy with your $20.00.

You can figure this out by writing an inequality to model the information given in the story.

First, write down what you know from the given information.

A bag of chips costs $2.50 and a bottle of soda costs $1.50. The number of bags of chips you buy and the number of bottles of sodas must be less than or equal to $20.00.

Next, name the variables for the inequality.

Let ‘\begin{align*}c\end{align*}c’ = the number of bags of chips

Let ‘\begin{align*}s\end{align*}s’ = the number of sodas

Next, write the inequality using the two variables.

\begin{align*}1.50s+2.50c \le 20.00\end{align*}1.50s+2.50c20.00

Next, test the point \begin{align*}(1,1)\end{align*}(1,1). Substitute \begin{align*}x=1\end{align*}x=1 and \begin{align*}y=1\end{align*}y=1 into the given inequality and simplify.

\begin{align*}\begin{array}{rcl} 1.50s+2.50c & \le & 20.00 \\ 1.50(1)+2.50(1) & \le & 20.00 \\ 1.50+2.50 & \le & 20.00 \\ 4.00 & \le & 20.00 \end{array}\end{align*}1.50s+2.50c1.50(1)+2.50(1)1.50+2.504.0020.0020.0020.0020.00

The point \begin{align*}(1,1)\end{align*}(1,1) makes the inequality true. The location of the test point with respect to the boundary line will be shaded.

Next, graph the inequality using a solid line, on a Cartesian grid.

License: CC BY-NC 3.0

 sodas and 5 bags of chips.\begin{align*}(5,5) \rightarrow 5\end{align*}(5,5)5 sodas and 4 bags of chips; and \begin{align*}(6,4) \rightarrow 6\end{align*}(6,4)6 sodas and 2 bags of chips; \begin{align*}(9,2) \rightarrow 9\end{align*}(9,2)9With your $20.00 you can buy any combination of soda and chips shown in the shaded area of the graph. Some possible purchases you could make are

Example 2

Graph the solution set of the following inequality. 

\begin{align*}y \ge 2x-3\end{align*}y2x3

First, write down the information given by the inequality.

The \begin{align*}y\end{align*}y-intercept of the line is \begin{align*}(0,-3)\end{align*}(0,3) and the slope of the line is 2. The boundary line will be a solid line since the inequality sign is \begin{align*}\ge\end{align*}.

Next, test the point \begin{align*}(1,1)\end{align*}(1,1).

\begin{align*}\begin{array}{rcl} y & \ge & 2x-3 \\ 1 & \ge & 2(1)-3 \\ 1 & \ge & 2-3 \\ 1 & \ge & -1 \end{array} \end{align*}y1112x32(1)3231

The test point of \begin{align*}(1,1)\end{align*}(1,1) makes the inequality true. Therefore, the side of the boundary line containing the test point must be shaded.

License: CC BY-NC 3.0

Example 3

Does the test point \begin{align*}(1,1)\end{align*}(1,1) make the inequality \begin{align*}y \ge 3x-4\end{align*}y3x4 true?

First, substitute \begin{align*}x=1\end{align*}x=1 and \begin{align*}y =1\end{align*}y=1 into the given inequality.

\begin{align*} \begin{array}{rcl} y & \ge & 3x-4 \\ 1 & \ge & 3(1)-4 \end{array} \end{align*}y13x43(1)4 

Next, simplify the right side of the inequality.

\begin{align*} \begin{array}{rcl} 1 & \ge & 3(1)-4 \\ 1 & \ge & 3-4 \\ 1 & \ge & -1 \end{array} \end{align*}1113(1)4341

The number 1 is greater than the number -1. The test point makes the inequality true.

Example 4

What type of boundary line will be displayed on the graph of the inequality \begin{align*}y < \frac{2}{3}x-5\end{align*}y<23x5?

The inequality symbol displayed in the given inequality is < which means the boundary line will be a dashed or broken line.

Example 5

What does the shaded area on the graph of an inequality indicate?

The shaded area on the graph of an inequality indicates the location of the solution set for the inequality.


Write an inequality for each graph.


License: CC BY-NC 3.0

License: CC BY-NC 3.0

License: CC BY-NC 3.0

License: CC BY-NC 3.0

Graph the following inequalities on the coordinate plane.

5. \begin{align*}y < 2x + 1\end{align*}y<2x+1

6. \begin{align*}y \ge 3x - 2\end{align*}y3x2

7. \begin{align*}y \ge \frac{-1}{2}x\end{align*}y12x

8. \begin{align*}y \le \frac{1}{4}x + 2\end{align*}y14x+2

9. \begin{align*}y < - 2x\end{align*}y<2x

10. \begin{align*}y \le 4\end{align*}y4

Answer each question true or false.

11. You can’t shade less than a vertical line.

12. A dotted line can only be used in an inequality with greater than.

13. A dotted line is used when the inequality sign does not include an equals.

14. You can shade less than or greater than a horizontal line.

15. You can graph a linear inequality in two variables.

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.17. 

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Cartesian Plane The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.
inequality An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are <, >, \le, \ge and \ne.
Linear Inequality Linear inequalities are inequalities that can be written in one of the following four forms: ax + b > c, ax + b < c, ax + b \ge c, or ax + b \le c.
Slope-Intercept Form The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.

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