Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 3.2 - 1

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 3.2 - 1


Linear Inequalities and Absolute Value

Chapter 3


3.2

Set Operations and Compound

Inequalities

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.2 - 4

3.2 Set Operations and Compound Inequalities

Objectives

1. Find the intersection of two sets.

2. Solve compound inequalities with

the word and.

3. Find the union of two sets.

4. Solve compound inequalities with

the word or.



Find the Intersection of Two sets

The intersection of two sets is defined using the word and.

Intersection of Sets

A B



Find the Intersection of Two Sets

–3 –1

04 5

3



Compound Inequalities

A compound inequality is a statement made by combining two inequalities with the words and or or.

Examples of Compound Inequalities



Solving Compound Inequalities

To solve compound inequalities with the word and, use the following steps.

Step 1 Solve each inequality individually.

Step 2 Since the inequalities are joined with and, the solution set of the compound inequality will

include all numbers that satisfy both inequalities in Step 1 (the intersection of the solution sets).




Solve the compound inequality:






Step 2 Because the inequalities are joined by the word and, the solution will include all numbers that satisfy both inequalities in Step 1. The graphs of the two inequalities are:

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5





Step 2 Find the intersection of the two graphs.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5





Step 1 Solve each inequality in the compound inequality individually.






1 6–2 –1 0 2 3 4 5 7 86–2 –1 0 1 2 3 4 5 7 8

1 6–2 –1 0 2 3 4 5 7 86–2 –1 0 1 2 3 4 5 7 8





Step 2

1 6–2 –1 0 2 3 4 5 7 86–2 –1 0 1 2 3 4 5 7 8

1 6–2 –1 0 2 3 4 5 7 86–2 –1 0 1 2 3 4 5 7 8

Find the intersection of the two graphs.

1 6–2 –1 0 2 3 4 5 7 86–2 –1 0 1 2 3 4 5 7 8











–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5





Step 2 Find the intersection of the two graphs.

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5

The intervals do not overlap.



Find the Union of Two sets

The union of two sets is defined using the word or.

Union of Sets

A B



Find the Union of Two Sets

–3 –1

04 5

3




To solve compound inequalities with the word or, use the following steps.


Step 2 Since the inequalities are joined with or, the solution set includes all numbers that satisfy either one of the two inequalities in Step 1 (the union of the solution sets).









The graphs of these two inequalities are:

–10 –8 –6 –4 –2 0 2 4 6 8 10

–10 –8 –6 –4 –2 0 2 4 6 8 10





Step 2 Because the inequalities are joined by the word or, find the union of the two solution sets.

–10 –8 –6 –4 –2 0 2 4 6 8 10

It would be wrong to write since it implies that .









The graphs of these two inequalities are:

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5





Step 2 Because the inequalities are joined by the word or, find the union of the two solution sets.

–10 –8 –6 –4 –2 0 2 4 6 8 10

By taking the union, we obtain every real number as a solution set.

Documents

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 3.2 - 1