Section 3.7Graphing Linear Inequalities

Objectives

Determine whether an ordered pair is a solution of an inequality

Graph a linear inequality in two variables Graph inequalities with a boundary

through the origin Solve applied problems involving linear

inequalities in two variables

Objective 1: Determine Whether an Ordered Pair Is a Solution of an Inequality

Recall that an inequality is a statement that contains one of the symbols <, ≤, >, or ≥. Inequalities in one variable, such as x + 6 < 8 and

5x + 3 ≥ 4x, were solved in Section 2.7.Because they have an infinite number of solutions, we represented their solution sets graphically, by shading intervals on a number line.

We now extend that concept to linear inequalities in two variables, as we introduce a procedure that is used to graph their solution sets.

Objective 1: Determine Whether an Ordered Pair Is a Solution of an Inequality

If the = symbol in a linear equation in two variables is replaced with an inequality symbol, we have a linear inequality in two variables. Some examples of linear inequalities in two variables

are: x − y ≤ 5, 4x + 3y < −6, and y > 2x and x < –3 As with linear equations, an ordered pair (x, y)

is a solution of an inequality in x and y if a true statement results when the values of the variables are substituted into the inequality.

EXAMPLE 1 Determine whether each ordered pair is a

solution of x − y ≤ 5. Then graph each solution:

a. (4, 2) b. (0, −6) c. (1, −4)

Objective 2: Graph a Linear Inequality in Two Variables In Example 1, we graphed two of the solutions

of x − y ≤ 5. Since there are infinitely more ordered pairs (x, y)

that make the inequality true, it would not be reasonable to plot all of them. Fortunately, there is an easier way to show all of the solutions.

The graph of a linear inequality is a picture that represents the set of all points whose coordinates satisfy the inequality.

In general, such graphs are regions bounded by a line. We call those regions half-planes, and we use a two-step procedure to find them.

Objective 2: Graph a Linear Inequality in Two Variables Graphing Linear Inequalities in Two Variables:

Replace the inequality symbol with an equal symbol = and graph the boundary line of the region. If the original inequality allows the possibility of equality (the symbol is either ≤ or ≥), draw the boundary line as a solid line. If equality is not allowed (< or >), draw the boundary line as a dashed line.

Pick a test point that is on one side of the boundary line. (Use the origin if possible.) Replace x and y in the inequality with the coordinates of that point. If a true statement results, shade the side that contains that point. If a false statement results, shade the other side of the boundary.

EXAMPLE 3

Graph: 4x + 3y < −6

Objective 3: Graph Inequalities with a Boundary through the Origin In the next example, the boundary line

passes through the origin.

EXAMPLE 4

Graph: y > 2x.

Objective 4: Solve Applied Problems Involving Linear Inequalities in Two Variables

When solving applied problems, phrases such as at least, at most, and should not exceed indicate that an inequality should be used.

EXAMPLE 6 Carlos has two part-time jobs, one paying $10 per hour and

another paying $12 per hour. If x represents the number of hours he works on the first job, and y represents the number of hours he works on the second, the graph of 10x + 12y ≥ 240 shows the possible ways he can schedule his time to earn at least $240 per week to pay his college expenses. Find four possible combinations of hours he can work to achieve his financial goal.

Working Two Jobs