2. Linear Inequalities and Absolute Value
Chapter 3
Linear Inequalities and Absolute Value

3. Set Operations and Compound Inequalities
3.2
Set Operations and Compound
Inequalities

4. 3.2 Set Operations and Compound Inequalities
Objectives
Find the intersection of two sets.
Solve compound inequalities with
the word and.
Find the union of two sets.
4. Solve compound inequalities with
the word or.
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5. 3.2 Set Operations and Compound Inequalities
Find the Intersection of Two sets
The intersection of two sets is defined using the word and.
Intersection of Sets A B
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6. 3.2 Set Operations and Compound Inequalities
Find the Intersection of Two Sets –1 –3 3 4 5
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7. 3.2 Set Operations and Compound Inequalities
A compound inequality is a statement made by combining two inequalities with the words and or or.
Examples of Compound Inequalities
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8. 3.2 Set Operations and 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).
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9. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
Step 1
Solve each inequality individually.
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10. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
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 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5
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11. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
Step 2
Find the intersection of the two graphs. –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5
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12. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
Solve each inequality in the compound inequality individually.
Step 1
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13. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
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: 1 6 –2 –1 2 3 4 5 7 8 1 6 –2 –1 2 3 4 5 7 8
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14. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
Step 2
Find the intersection of the two graphs. 6 –2 –1 2 3 4 5 7 8 –2 –1 1 1 2 3 4 5 6 7 8 1 6 –2 –1 2 3 4 5 7 8 1 6 –2 –1 2 3 4 5 7 8
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15. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
Solve each inequality in the compound inequality individually.
Step 1
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16. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
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 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5
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17. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
Step 2
Find the intersection of the two graphs. –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5
The intervals do not overlap. –5 –4 –3 –2 –1 1 2 3 4 5
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18. 3.2 Set Operations and Compound Inequalities
Find the Union of Two sets
The union of two sets is defined using the word or.
Union of Sets A B
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19. 3.2 Set Operations and Compound Inequalities
Find the Union of Two Sets –1 –3 3 4 5
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20. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
To solve compound inequalities with the word or, use the following steps.
Step 1 Solve each inequality individually.
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).
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21. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
Solve each inequality individually.
Step 1
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22. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
The graphs of these two inequalities are:
–10 –8 –6 –4 –2 2 4 6 8 10
–10 –8 –6 –4 –2 2 4 6 8 10
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23. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Step 2
Because the inequalities are joined by the word or, find the union of the two solution sets.
–10 –8 –6 –4 –2 2 4 6 8 10
It would be wrong to write since it implies that
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24. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
Solve each inequality in the compound inequality individually.
Step 1
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25. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Solve the compound inequality:
The graphs of these two inequalities are: –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5
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26. 3.2 Set Operations and Compound Inequalities
Solving Compound Inequalities
Step 2
Because the inequalities are joined by the word or, find the union of the two solution sets.
–10 –8 –6 –4 –2 2 4 6 8 10
By taking the union, we obtain every real number as a solution set.
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