Intermediate Algebra (11th edition)

SECTION 2.6 Set Operations and Compound Inequalities 105

Solving a Compound Inequality with and

Solve the compound inequality, and graph the solution set.

and

Step 1 Solve each inequality individually.

and

The graphs of x 63 and x 712 are shown in FIGURE 21.

x 63 x 712

x+ 265 x - 1072

x+ 265 x- 1072

NOW TRY EXAMPLE 4

EXERCISE 4
Solve and graph.

and 2 x+ 175

x- 7 6- 12

0 3 6 9

12

x < 3

x > 12

FIGURE 21

0 3 6 9 12

0

FIGURE 22

Union of Sets

For any two sets Aand B, the unionof Aand B, symbolized is defined as

follows.

A ́B= 5 x |x is an element of A or x is an element of B 6

A ́B,

AB

Step 2 There is no number that is both less than 3 andgreater than 12, so the given

compound inequality has no solution. The solution set is 0 .See FIGURE 22.

Finding the Union of Two Sets

Let and Find

Begin by listing all the elements of set A: 1, 2, 3, 4. Then list any additional

elements from set B. In this case the elements 2 and 4 are already listed, so the only

additional element is 6.

The union consists of all elements in either A or B(or both). NOW TRY

NOTE In Example 5,notice that although the elements 2 and 4 appeared in both sets

Aand B, they are written only once in A ́B.

= 5 1, 2, 3, 4, 6 6

A ́B= 5 1, 2, 3, 4 6 ́ 5 2, 4, 6 6

A= 5 1, 2, 3, 4 6 B= 5 2, 4, 6 6. A ́B.

EXAMPLE 5

NOW TRY

OBJECTIVE 3 Find the union of two sets. The union of two sets is defined

with the word or.

NOW TRY
EXERCISE 5
Let
and
FindA ́B.

B= 5 5, 15, 25 6.

A= 5 5, 10, 15, 20 6

NOW TRY ANSWERS

0 5. 5 5, 10, 15, 20, 25 6