Section P9 Linear Inequalities and Absolute Value Inequalities

Interval Notation

Example Express the interval in set builder notation and graph:

  3, 2    , 2 

Intersections and Unions of Intervals

Example Find the set:

  



Example Find the set:

  



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

x x

7

Example Solve the inequality:

x

 2 1

x

3 3 

x

2  2 

Checking the solution of a linear inequality on a Graphing Calculator 2

x x

4

Separate the inequality into two equations

.

y

1  2

x

 1

y

2 4

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 y 1 is greater than y 2 .

Y 2 =-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.

Y 1 =2x+1

  . Let's look at these unusual solutions.

x

1

x

Never true  1 Always true  

Example Solve the inequality:

x x

4

Solving Compound Inequalities

Now consider two inequalities such as express as a compound inequality 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.

x

1 2 y     x             

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 

c

will be a single interval. Avoid the common error of rewriting x 

c

as -cc.

Example Solve and graph the solution set on a number line.

2

x

3

Example Solve and graph the solution set on a number line.

2

x

3

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?

Solve the absolute value inequality.

2

x

4 (a) (b) (c) (d)

x

0

x

  4 or x

x

  4 and x  0  0 x  4 or x  0

Solve the linear inequality.

4 

x

 6

x

 9 (a)

x

  3 2 (b) (c) (d)

x x

  3   3 2

x

 2