Section 1.7Linear Inequalities

andAbsolute Value Inequalities

Interval Notation

Example

Express the interval in set builder notation and graph:

3,2

0,4

,2

Intersections and

Unions of Intervals

Example

Find the set: 2,3 0,4

Example

Find the set: 2,3 0,4

Solving Linear Inequalities

in One Variable

A linear inequality in x can be written in one of the following

forms: ax+b<0, ax+b 0, ax+b>0 ax+b 0. In each form

a 0.

Example:

-x+7 0

-x -7

x 7

When we multiply or divide b

oth sides of an inequality by

a negative number, the direction of the inequality symbol

is reversed.

Example

Solve and graph the solution set on a number line:

4 5 7x x

Checking the solution of a linear inequality on a Graphing Calculator

1

2

2 1

4

y x

y x

2 1 4x x

Y1=2x+1

Y2=-x+4

The region on the graph of the red box is where y1 is greater than y2. This is when x is greater than 1.

The intersection of the two lines is at (1,3). You can see this because both y values are the same, – 3.

The region in the red box is where the values of y1 is greater than y2.

Separate the inequality into two equations.

Inequalities with

Unusual Solution Sets

The solution set could be the null set, . The solution

set could be all real numbers, - , .

1

0 1

Never true

x x

1

0 1

Always true ,

x x

Example

Solve each inequality:

3 4x x

Solving

Compound Inequalities

Now consider two inequalities such as

-3<2x+1 and 2x+1 3

express as a compound inequality

-3<2x+1 3

In this shorter form we can solve both inequalities

at once by performing the same operation on all t

hree

parts of the inequality. The goal is to isolate the x in

the middle.

Example

Solve and graph the solution set on a number line.

3 1 2x

x

y

Solving Inequalities

with Absolute Value

The graph of the solution set for x >c will be divided

into two intervals whose union cannot be represented as

a single interval. The graph of the solution set for x

will be a single interval. Avoid

c

the common error of rewriting

x as -c<x>c.c

Example

Solve and graph the solution set on a number line.

2 5 3x

Example

Solve and graph the solution set on a number line.

2 5 3x

Applications

Example

A national car rental company charges a flat rate of $320 per week for the rental of a 4 passenger sedan. The same car can be rented from a local car rental company which charges $180 plus $ .20 per mile. How many miles must be driven in a week to make the rental cost for the national company a better deal than the local company?

(a)

(b)

(c)

(d)

2 2 4x

Solve the absolute value inequality.

4 0

4 or x 0

4 and x 0

x 4 or x 0

x

x

x

(a)

(b)

(c)

(d)

4 3 6 9x x Solve the linear inequality.

3

23

23

2

x

x

x

x