Solutions to Try Its

1. [latex]|x - 2|\le 3\\[/latex]

2. using the variable [latex]p\\[/latex] for passing, [latex]|p - 80|\le 20\\[/latex]

3. [latex]f\left(x\right)=-|x+2|+3\\[/latex]

4. [latex]x=-1\\[/latex] or [latex]x=2\\[/latex]

5. [latex]f\left(0\right)=1\\[/latex], so the graph intersects the vertical axis at [latex]\left(0,1\right)\\[/latex]. [latex]f\left(x\right)=0\\[/latex] when [latex]x=-5\\[/latex] and [latex]x=1\\[/latex] so the graph intersects the horizontal axis at [latex]\left(-5,0\right)\\[/latex] and [latex]\left(1,0\right)\\[/latex].

6. [latex]4\le x\le 8\\[/latex]

7. [latex]k\le 1\\[/latex] or [latex]k\ge 7\\[/latex]; in interval notation, this would be [latex]\left(-\infty ,1\right]\cup \left[7,\infty \right)\\[/latex]

Solutions to Odd-Numbered Exercises

1. Isolate the absolute value term so that the equation is of the form [latex]|A|=B\\[/latex]. Form one equation by setting the expression inside the absolute value symbol, [latex]A\\[/latex], equal to the expression on the other side of the equation, [latex]B\\[/latex]. Form a second equation by setting [latex]A\\[/latex] equal to the opposite of the expression on the other side of the equation, -B. Solve each equation for the variable.

3. The graph of the absolute value function does not cross the [latex]x\\[/latex] -axis, so the graph is either completely above or completely below the [latex]x\\[/latex] -axis.

5. First determine the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to determine which values satisfy the inequality.

7. [latex]|x+4|=\frac{1}{2}\\[/latex]

9. [latex]|f\left(x\right)-8|<0.03\\[/latex]

11. [latex]\left\{1,11\right\}\\[/latex]

13. [latex]\left\{\frac{9}{4},\frac{13}{4}\right\}\\[/latex]

15. [latex]\left\{\frac{10}{3},\frac{20}{3}\right\}\\[/latex]

17. [latex]\left\{\frac{11}{5},\frac{29}{5}\right\}\\[/latex]

19. [latex]\left\{\frac{5}{2},\frac{7}{2}\right\}\\[/latex]

21. No solution

23. [latex]\left\{-57,27\right\}\\[/latex]

25. [latex]\left(0,-8\right);\left(-6,0\right),\left(4,0\right)\\[/latex]

27. [latex]\left(0,-7\right)\\[/latex]; no [latex]x\\[/latex] -intercepts

29. [latex]\left(-\infty ,-8\right)\cup \left(12,\infty \right)\\[/latex]

31. [latex]\frac{-4}{3}\le x\le 4\\[/latex]

33. [latex]\left(-\infty ,-\frac{8}{3}\right]\cup \left[6,\infty \right)\\[/latex]

35. [latex]\left(-\infty ,-\frac{8}{3}\right]\cup \left[16,\infty \right)\\[/latex]

37.

39.

41.

43.

45.

47.

49.

51.

53. range: [latex]\left[0,20\right][/latex]

55. [latex]x\text{-}[/latex] intercepts:

57. [latex]\left(-\infty ,\infty \right)[/latex]

59. There is no solution for [latex]a[/latex] that will keep the function from having a [latex]y[/latex] -intercept. The absolute value function always crosses the [latex]y[/latex] -intercept when [latex]x=0[/latex].

61. [latex]|p - 0.08|\le 0.015[/latex]

63. [latex]|x - 5.0|\le 0.01[/latex]