Section Exercises

1. How do you solve an absolute value equation?

2. How can you tell whether an absolute value function has two x-intercepts without graphing the function?

3. When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

4. How can you use the graph of an absolute value function to determine the x-values for which the function values are negative?

5. How do you solve an absolute value inequality algebraically?

6. Describe all numbers $x$ that are at a distance of 4 from the number 8. Express this using absolute value notation.

7. Describe all numbers $x$ that are at a distance of $\frac{1}{2}$ from the number −4. Express this using absolute value notation.

8. Describe the situation in which the distance that point $x$ is from 10 is at least 15 units. Express this using absolute value notation.

9. Find all function values $f\left(x\right)$ such that the distance from $f\left(x\right)$ to the value 8 is less than 0.03 units. Express this using absolute value notation.

For the following exercises, solve the equations below and express the answer using set notation.

10. $|x+3|=9$

11. $|6-x|=5$

12. $|5x - 2|=11$

13. $|4x - 2|=11$

14. $2|4-x|=7$

15. $3|5-x|=5$

16. $3|x+1|-4=5$

17. $5|x - 4|-7=2$

18. $0=-|x - 3|+2$

19. $2|x - 3|+1=2$

20. $|3x - 2|=7$

21. $|3x - 2|=-7$

22. $\left|\frac{1}{2}x - 5\right|=11$

23. $\left|\frac{1}{3}x+5\right|=14$

24. $-\left|\frac{1}{3}x+5\right|+14=0$

For the following exercises, find the x- and y-intercepts of the graphs of each function.

25. $f\left(x\right)=2|x+1|-10$

26. $f\left(x\right)=4|x - 3|+4$

27. $f\left(x\right)=-3|x - 2|-1$

28. $f\left(x\right)=-2|x+1|+6$

For the following exercises, solve each inequality and write the solution in interval notation.

29. $\left|x - 2\right|>10$

30. $2|v - 7|-4\ge 42$

31. $|3x - 4|\le 8$

32. $|x - 4|\ge 8$

33. $|3x - 5|\ge 13$

34. $|3x - 5|\ge -13$

35. $\left|\frac{3}{4}x - 5\right|\ge 7$

36. $\left|\frac{3}{4}x - 5\right|+1\le 16$

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

37. $y=|x - 1|$

38. $y=|x+1|$

39. $y=|x|+1$

For the following exercises, graph the given functions by hand.

40. $y=|x|-2$

41. $y=-|x|$

42. $y=-|x|-2$

43. $y=-|x - 3|-2$

44. $f\left(x\right)=-|x - 1|-2$

45. $f\left(x\right)=-|x+3|+4$

46. $f\left(x\right)=2|x+3|+1$

47. $f\left(x\right)=3|x - 2|+3$

48. $f\left(x\right)=|2x - 4|-3$

49. $f\left(x\right)=|3x+9|+2$

50. $f\left(x\right)=-|x - 1|-3$

51. $f\left(x\right)=-|x+4|-3$

52. $f\left(x\right)=\frac{1}{2}\left|x+4\right|-3$

53. Use a graphing utility to graph $f\left(x\right)=10|x - 2|$ on the viewing window $\left[0,4\right]$. Identify the corresponding range. Show the graph.

54. Use a graphing utility to graph $f\left(x\right)=-100|x|+100$ on the viewing window $\left[-5,5\right]$. Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

55. $f\left(x\right)=\left(-0.1\right)\left|0.1\left(0.2-x\right)\right|+0.3$

56. $f\left(x\right)=4\times {10}^{9}\left|x-\left(5\times {10}^{9}\right)\right|+2\times {10}^{9}$

For the following exercises, solve the inequality.

57. $\left|-2x-\frac{2}{3}\left(x+1\right)\right|+3>-1$

58. If possible, find all values of $a$ such that there are no $x\text{-}$ intercepts for $f\left(x\right)=2|x+1|+a$.

59. If possible, find all values of $a$ such that there are no $y$ -intercepts for $f\left(x\right)=2|x+1|+a$.

60. Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and $x$ represents the distance from city B to city A, express this using absolute value notation.

61. The true proportion $p$ of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

62. Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable $x$ for the score.

63. A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using $x$ as the diameter of the bearing, write this statement using absolute value notation.

64. The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is $x$ inches, express the tolerance using absolute value notation.