Solutions 6: Absolute Value Functions

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]

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