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Section 7.4, Slide *

CHAPTER 7

Algebra: Graphs, Functions, and Linear Systems

Section 7.4, Slide *

7.4

Linear Inequalities in Two Variables

Section 7.4, Slide *

Objectives

Graph a linear inequality in two variables.
Use mathematical models involving linear inequalities.
Graph a system of linear inequalities.

Section 7.4, Slide *

Linear Inequalities in Two Variables and Their Solutions

If we change the symbol = in the equation Ax + By = C to >, <, ≥, or ≤, we obtain a linear inequality in two variables.

For example, x + y < 2 and 3x – 5y ≥ 15 are linear inequalities in two variables.

A solution of an inequality in two variables, x and y, is an ordered pair of real numbers such that when the x-coordinate is substituted for x and the y-coordinate is substituted for y in the inequality and we obtain a true statement.

Section 7.4, Slide *

The Graph of a Linear Inequality in Two Variables

The graph of an inequality in two variables is the set of all points whose coordinates satisfy the inequality.

Section 7.4, Slide *

Example: Graphing a Linear Inequality in Two Variables

Graph: 3x – 5y ≥ 15.

Solution:

Step 1 We need to graph 3x – 5y = 15. We can use intercepts to graph this line.

We set y = 0 to We set x = 0 to

find the x-intercept. find the y-intercept.

3x – 5y = 15 3x – 5y = 15

3x – 5 · 0 = 15 3 · 0 – 5y = 15

3x = 15 −5y = 15

x = 5 y = −3

Section 7.4, Slide *

Example continued

The x-intercept is 5, so the line passes

through (5,0). The y-intercept is −3, so

the line passes through (0,−3).

Step 2 We choose (0,0) as a test point.

3x – 5y ≥ 15

3 · 0 – 5 · 0 ≥ 15

0 – 0 ≥ 15

0 ≥ 15 NOT TRUE!

Section 7.4, Slide *

Example continued

Step 3 Since the statement is false, we shade the half-plane that does not include the test point (0,0).

Thus, the graph with the shading is the solution to the given inequality.

Section 7.4, Slide *

Example: The Graph of a Linear Inequality in Two Variables

Graph:

Solution:

Step 1 We need to graph

Since the inequality > is given, we use a dashed line.

Section 7.4, Slide *

Example continued

Step 2 We choose a test point not on the line, (1, 1), which lies in the half-plane above the line.

TRUE!

Step 3 Since the statement is true, then

we shade the half-plane that includes

the test point (1, 1).

Section 7.4, Slide *

Graphing Linear Inequalities without Using Test Points

For the vertical line x = a:

If x > a, shade the half-plane to the right of x = a.

If x < a, shade the half-plane to the left of x = a.

For the horizontal line y = b:

If y > b, shade the half-plane above y = b.
If y < b, shade the half-plane below y = b.

Section 7.4, Slide *

Example: Graphing Inequalities Without Using Test Points

Graph each inequality in a rectangular coordinate system: a. y ≤ −3 b. x > 2

Solution: a. y ≤ −3 b. x > 2

Section 7.4, Slide *

Modeling with Systems of Linear Inequalities

Just as two or more linear equations make up a system of linear equations, two or more linear inequalities make up a system of linear inequalities. A solution of a system of linear inequalities in two variables is an ordered pair that satisfies each inequalities in the system.

Section 7.4, Slide *

Graphing Systems of Linear Inequalities

The solution set of a system of linear inequalities in two variables is the set of all ordered pairs that satisfy each inequality in the system.

Section 7.4, Slide *

Example: Graphing a System of Linear Inequalities

Graph the solution set of the system:

x – y < 1

2x + 3y ≥ 12.

Solution: Replacing each inequality symbol with an equal sign indicates that we need to graph x – y = 1 and 2x + 3y = 12. We can use intercepts to graph these lines.

Section 7.4, Slide *

Example continued

Section 7.4, Slide *

Example continued

Now we are ready to graph the solution set of the system of linear inequalities.

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