Problem 5 Writing Solutions of an Inequality

What are the solutions of | 2 x − 1 | < 3 ? vertical line 2 x minus 1 vertical line less than 3 question mark  Write the solutions as either the union or the intersection of two sets.

| 2 x − 1 | < 3   − 3 < 2 x − 1 < 3 − 2 < 2 x < 4 − 1 < x < 2 Write a compound inequality . Add  1  to each expression . Divide each side by  2. table with 2 rows and 2 columns , row1 column 1 , vertical line 2 x minus 1 vertical line less than 3 , column 2 , row2 column 1 , table with 3 rows and 3 columns , row1 column 1 , negative 3 less than , column 2 2 x minus 1 , column 3 less than 3 , row2 column 1 , negative 2 less than , column 2 2 x , column 3 less than 4 , row3 column 1 , negative 1 less than , column 2 x , column 3 less than 2 , end table , column 2 table with 3 rows and 1 column , row1 column 1 , cap writeacompoundinequality . . , row2 column 1 , cap add , 1 . toeachexpression . . , row3 column 1 , cap divideeachsideby . 2. , end table , end table

The solutions of the inequality are given by − 1 < x < 2 . negative 1 less than x less than 2 .  You can write this as x > − 1 x greater than negative 1  and x < 2 . x less than 2 .  This compound inequality is the intersection of two sets, which you can write as follows: { x | x > − 1 } ∩ { x | x < 2 } . left brace x vertical line x greater than negative 1 right brace intersection left brace x vertical line x less than 2 right brace .

✓ Got It?

5. Solve each inequality. Write the solutions as either the union or the intersection of two sets.
   1. 8 ≤ x + 5 < 11 8 less than or equal to x plus 5 less than 11
   2. | 4 x − 6 | > 14 vertical line 4 x minus 6 vertical line greater than 14

✓ Lesson Check

Do you know HOW?

Let X = { 2 , 4 , 6 , 8 , 10 } , Y = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } , x equals left brace 2 comma 4 comma 6 comma 8 comma 10 right brace comma y equals left brace 1 comma 2 comma 3 comma 4 comma 5 comma 6 comma 7 comma 8 comma 9 comma 10 right brace comma  and Z = { 1 , 3 , 5 , 7 , 9 } . z equals left brace 1 comma 3 comma 5 comma 7 comma 9 right brace .  Find each union or intersection.

1. X ∪ Y x union y
2. X ∩ Y x intersection y
3. X ∩ Z x intersection z
4. Y ∪ Z y union z
5. In a survey of 80 people who use their cell phones to take pictures and play games, 49 take pictures and 35 take pictures and play games. How many people only use their cell phones to play games?

Do you UNDERSTAND?

6. Vocabulary Suppose A and B are nonempty sets. Which set contains more elements: A ∪ B eh union b  or A ∩ B ? eh intersection b question mark  Explain your reasoning.
7. Compare and Contrast How are unions and intersections of sets different?

Determine whether each statement is true or false.

8. If x is an element of set A and x is not an element of set B, then x is an element of A ∪ B . eh union b .
9. If x is not an element of set A and x is an element of set B, then x is an element of A ∩ B . eh intersection b .