College Algebra, 2013a

2.2 Absolute Value Functions 183
2.2.1 Exercises
In Exercises 1 - 15, solve the equation.
1. |x| =6 2. |3x − 1| =10 3. |4 − x| =7
4. 4 −|x| = 3 5. 2|5x +1|−3=0 6. |7x − 1| +2=0
7.
5 −|x|
2
2
=1 8.
3 |5 − 2x|− 1 2
=5 9. |x| = x +3
10. |2x − 1| = x + 1 11. 4 −|x| =2x + 1 12. |x − 4| = x − 5
13. |x| = x 2 14. |x| =12− x 2 15. |x 2 − 1| =3
Prove that if |f(x)| = |g(x)| then either f(x) =g(x) orf(x) =−g(x). Use that result to solve the
equations in Exercises 16 - 21.
16. |3x − 2| = |2x +7| 17. |3x +1| = |4x| 18. |1 − 2x| = |x +1|
19. |4 − x|−|x +2| = 0 20. |2 − 5x| =5|x +1| 21. 3|x − 1| =2|x +1|
In Exercises 22 - 33, graph the function. Find the zeros of each function and the x- andy-intercepts
of each graph, if any exist. From the graph, determine the domain and range of each function, list
the intervals on which the function is increasing, decreasing or constant, and find the relative and
absolute extrema, if they exist.
22. f(x) =|x +4| 23. f(x) =|x| + 4 24. f(x) =|4x|
25. f(x) =−3|x| 26. f(x) =3|x +4|−4 27. f(x) = 1 |2x − 1|
3
28. f(x) =
|x +4|
x +4
29. f(x) =
|2 − x|
2 − x
30. f(x) =x + |x|−3
31. f(x) =|x +2|−x 32. f(x) =|x +2|−|x| 33. f(x) =|x +4| + |x − 2|
34. With the help of your classmates, find an absolute value function whose graph is given below.
4
3
2
1
y
−8−7−6−5−4−3−2−1 1 2 3 4 5 6 7 8
x
35. With help from your classmates, prove the second, third and fifth parts of Theorem 2.1.
36. Prove The Triangle Inequality: For all real numbers a and b, |a + b| ≤|a| + |b|.