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Graphing Linear Inequalities Study Guide

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Updated on Oct 3, 2011

Introduction to Graphing Linear Inequalities

A line is a dot that went for a walk.

–Paul Klee (1879–1940) Swiss Painter

In this lesson, you'll learn how to graph inequalities with two variables and how to graph the solution set of a system of inequalities.

The solutions of the equation of a line are each point on the graph of the line. For the line y = x – 1, the point (2,1) is on the line, and the point (2,1) is a solution of the equation. An inequality can have many more solutions than the equation of a line. The inequality y < x + 5 is true whenever y is less than five more than x. When x is 3, y can be any number that is less than 8. When x is 4, y can be any number that is less than 9. To show the solution of an inequality, we graph a line and then shade the side of the line where all of the solutions lie.

In order to graph y < x + 5, we must first graph the line y = x + 5. The point (1,6) is on that line, but is (1,6) a solution of y < x + 5? It is not, because 6 is not less than 1 + 5. The points on the line y = x + 5 are not part of the solution set of y = x + 5, so we graph a dashed line instead of a solid line.

Graphing Inequalities

Tip:

When an inequality contains the < or > symbol, the graph of the line will be dashed. When an inequality contains the ≤ or ≥ symbol, the graph of the line will be solid.

Which side of the line contains the solutions to y < x + 5? We pick a test point to decide which side of the line to shade. The point (0,0) is the best point to choose, because it makes our calculations easy. Substitute 0 for y and 0 for x:

      y < x + 5
      0 < 0 + 5 ?
      0 < 5

Because 0 is less than 5, the point (0,0) is part of the solution set. Shade the area below the line to show that this is where the solutions lie.

Graphing Inequalities

We can also graph inequalities with one variable. To graph y ≥ –4, we start by graphing the line y = –4. Every value on the line has a y value of –4, and because –4 ≤ –4, the points on this line are part of the solution set. We will make the graph of the line solid. Next, test the point (0,0) to see if it is part of the solution set. Because 0 is greater than or equal to –4, (0,0) is in the solution set. Shade the area above the line.

Graphing Inequalities

Graphing the Solution Set of a System of Inequalities

In Lessons 18 and 19, we learned two methods for solving a system of equations. However, substitution and elimination will not work for solving a system of inequalities, because the value of each variable is a set of numbers. Even if we could isolate a variable by itself, we could never substitute a single value for it in order to find the value of the other variables. We must use a graph to show the solution of a system of inequalities.

If we are given two inequalities, we can plot them both and shade the solution of each on the same set of axes. Our solution to the system is the overlapping area. If the two inequalities have no overlapping area, then the system has no solution.

Look at the following system of inequalities.

      y < 5x
      y > –x – 1

To find the solution to this system, we begin by graphing y = 5x and y = –x – 1. Both lines will be dashed since the inequality symbols are less than and greater than. The test point (0,0) cannot be used on the first inequality, because that point is on the line y = 5x. Try (1,1): Because 1 is less than 5, (1,1) is part of the solution to y < 5x. Shade the area to the right of the line. We can test y > –x – 1 with the point (0,0). Because 0 is greater than –1, the point (0,0) is part of the solution to y > –x – 1. Shade the area above that line. The overlapping area is in quadrants I and IV. This darker region is the solution to the set of inequalities.

Graphing Inequalities

Tip:

Sometimes, the only difference between two inequalities in a system is the inequality symbol, as in the system y < x + 1 and y > x + 1. This system has no solution, because there are no y values that can be less than x + 1 and greater than x + 1. If the two inequalities were yx + 1 and yx + 1, the solution would be all the values that are on the line y = x + 1, since that is the only area the two inequalities have in common. To sum up, when the only difference between two inequalities is the inequality symbol, if the symbols are ≤ and ≥, then the solution is all points on the graphed line. If the symbols are < and >, then the system has no solution.

Find practice problems and solutions for these concepts at Graphing Linear Inequalities Practice Questions.

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