The Holy Spirit and His Work

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2.6

Linear Inequalities in Two

Variables

What you should learn

GOAL 1 Graph linear

inequalities in two variables.

GOAL 2 Use linear

inequalities to solve real-life

problems, such as finding the

number of minutes you can

call relatives using a calling

card in Example 4.

Why you should learn it

RE

GRAPHING LINEAR INEQUALITIES

A linear inequality in two variables is an inequality that can be written in one of

the following forms:

Ax + By < C,

Ax + By ≤ C,

Ax + By > C,

Ax + By ≥ C

An ordered pair (x, y) is a solution of a linear inequality if the inequality is true

when the values of x and y are substituted into the inequality. For instance, (º6, 2)

is a solution of y ≥ 3x º 9 because 2 ≥ 3(º6) º 9 is a true statement.

EXAMPLE 1

Checking Solutions of Inequalities

Check whether the given ordered pair is a solution of 2x + 3y ≥ 5.

a. (0, 1)

b. (4, º1)

c. (2, 1)

E

FE

᭢ To model real-life data,

such as blood pressures in

your arm and ankle

in Ex. 45.

AL LI

GOAL 1

SOLUTION

ORDERED PAIR

SUBSTITUTE

CONCLUSION

a. (0, 1)

2(0) + 3(1) = 3 ≥ 5

/

(0, 1) is not a solution.

b. (4, º1)

2(4) + 3(º1) = 5 ≥ 5

(4, º1) is a solution.

c. (2, 1)

2(2) + 3(1) = 7 ≥ 5

(2, 1) is a solution.

ACTIVITY

Developing

Concepts

Investigating the Graph of an Inequality

1

Copy the scatter plot.

2

Test each circled point to see whether it is a

solution of x + y ≥ 1. If it is a solution, color

it blue. If it is not a solution, color it red.

3

Graph the line x + y = 1. What relationship

do you see between the colored points and

the line?

4

y

1

Describe a general strategy for graphing an

inequality in two variables.

1

The graph of a linear inequality in two variables is the graph of all solutions of the

inequality. The boundary line of the inequality divides the coordinate plane into two

half-planes: a shaded region which contains the points that are solutions of the

inequality, and an unshaded region which contains the points that are not.

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