Check Your Skills Answer Key

1. 

2. 

3. 

4. z > v

5. Let a = total amount.

   • a > $2,000

6. (A), (B), (C), (D)

   All of these numbers are to the left of 10 on the number line.

   

7. x − 6 < 13
              x < 19

8. y + 11 ≥ −13

            y ≥ −24

9. x + 7 > 7

             x > 0

10. • x + 3 ≥ −2

             x ≥ −5

11. • −2y < 8

        y > −4

12. • a + 4 ≥ 2a
             4 ≥ a

13. True

14. False (Note that absolute value is always positive!)

15. False

16. True

17. True (|3 − 6| = |−3| = 3)

18. False

19. |a| = 6
    a = 6	or	a = −6

20. x = 3 or −7

    |x + 2| = 5
    + (x + 2) = 5	or	−(x + 2) = 5
    x + 2 = 5	or	−x − 2 = 5
    x = 3	or	−x = 7
    		x = −7

21. y = 7 or 

    |3y − 4| = 17
    + (3y − 4) = 17	or	−(3y − 4) = 17
    3y − 4 = 17	or	−3y + 4 = 17
    3y = 21	or	−3y = 13
    y = 7	or	y =

22. x = 4 or −5

    or 4|x + | = 18
    + (x + ) = 4	or	−(x + ) = 4
    x + = 4	or	−x − = 4
    x = 4	or	−x = 5
    		x = −5

23. |x + 1| > 2
    + (x + 1) > 2	or	−(x + 1) > 2
    x + 1 > 2	or	−x − 1 > 2
    x > 1	or	−x > 3
    		x < −3

    

24. |−x − 4| ≥ 8
    + (−x − 4) ≥ 8	or	−(−x − 4) ≥ 8
    −x − 4 ≥ 8	or	x + 4 ≥ 8
    −x ≥ 12	or	x ≥ 4
    x ≤ −12

    

25. |x − 7| < 9
    + (x − 7) < 9	and	−(x − 7) < 9
    x − 7 < 9	and	−x + 7 < 9
    x < 16	and	−x < 2
    		x > −2

    

    x > −2

    and

    x < 16,  −2 < x < 16

26. −3 < x < 5

    −7 < 3 − 2x < 9
    −10 < −2x < 6	Subtract 3 from all three terms.
    5 > x > −3	Divide all three terms by −2, and flip the inequality signs.
    or −3 < x < 5

27. (E)

    Extreme values for a are greater than −4 and less than 4. Extreme values for b are greater than −2 and less than −1.

    Note that a can be positive, zero, or negative, while b can only be negative, so ab can be positive, zero, and negative.

    The most negative ab can be is (less positive than 4) × (less negative than −2) = less negative than −8.

    The most positive ab can be is (less negative than −4) × (less negative than −2) = less positive than 8.

28. −10

    a	b	b − a
    −1	−6	−6 − (−1) = −5
    −1	−2	−2 − (−1) = −1
    4	−6	−6 − 4 = −10
    4	−2	−2 − 4 = −6

29. 2

    (x + 2)2 ≤ 2 − y
    y + (x + 2)2 ≤ 2	Add y to both sides.
    y ≤ 2 − (x + 2)2	Subtract (x + 2)2 from both sides.

    Note that y is maximized when (x + 2)² is minimized. The smallest possible value for (x + 2)² is 0, when x = −2. When (x + 2)² = 0, y = 2.