{"id":1021,"date":"2015-11-12T18:35:32","date_gmt":"2015-11-12T18:35:32","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1021"},"modified":"2015-11-12T18:35:32","modified_gmt":"2015-11-12T18:35:32","slug":"solutions-51","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/solutions-51\/","title":{"raw":"Solutions","rendered":"Solutions"},"content":{"raw":"

Solutions to Try Its<\/h2>\n

1.\u00a0[latex]|x - 2|\\le 3\\\\[\/latex]<\/span><\/p>\n

2.\u00a0using the variable [latex]p\\\\[\/latex] for passing, [latex]|p - 80|\\le 20\\\\[\/latex]<\/span><\/p>\n

3.\u00a0[latex]f\\left(x\\right)=-|x+2|+3\\\\[\/latex]<\/span><\/p>\n

4.\u00a0[latex]x=-1\\\\[\/latex] or [latex]x=2\\\\[\/latex]<\/span><\/p>\n

5.\u00a0[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].<\/span><\/p>\n

6.\u00a0[latex]4\\le x\\le 8\\\\[\/latex]<\/span><\/p>\n

7.\u00a0[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]<\/span><\/p>\n

Solutions to Odd-Numbered Exercises<\/b><\/span><\/p>\n

1.\u00a0Isolate 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.<\/span><\/p>\n

3.\u00a0The 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.<\/span><\/p>\n

5.\u00a0First 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.<\/span><\/p>\n

7.\u00a0[latex]|x+4|=\\frac{1}{2}\\\\[\/latex]<\/span><\/p>\n

9.\u00a0[latex]|f\\left(x\\right)-8|<0.03\\\\[\/latex]<\/span><\/p>\n

11.\u00a0[latex]\\left\\{1,11\\right\\}\\\\[\/latex]<\/span><\/p>\n

13.\u00a0[latex]\\left\\{\\frac{9}{4},\\frac{13}{4}\\right\\}\\\\[\/latex]<\/span><\/p>\n

15.\u00a0[latex]\\left\\{\\frac{10}{3},\\frac{20}{3}\\right\\}\\\\[\/latex]<\/span><\/p>\n

17.\u00a0[latex]\\left\\{\\frac{11}{5},\\frac{29}{5}\\right\\}\\\\[\/latex]<\/span><\/p>\n

19.\u00a0[latex]\\left\\{\\frac{5}{2},\\frac{7}{2}\\right\\}\\\\[\/latex]<\/span><\/p>\n

21.\u00a0No solution<\/span><\/p>\n

23.\u00a0[latex]\\left\\{-57,27\\right\\}\\\\[\/latex]<\/span><\/p>\n

25.\u00a0[latex]\\left(0,-8\\right);\\left(-6,0\\right),\\left(4,0\\right)\\\\[\/latex]<\/span><\/p>\n

27.\u00a0[latex]\\left(0,-7\\right)\\\\[\/latex]; no [latex]x\\\\[\/latex] -intercepts<\/span><\/p>\n

29.\u00a0[latex]\\left(-\\infty ,-8\\right)\\cup \\left(12,\\infty \\right)\\\\[\/latex]<\/span><\/p>\n

31.\u00a0[latex]\\frac{-4}{3}\\le x\\le 4\\\\[\/latex]<\/span><\/p>\n

33.\u00a0[latex]\\left(-\\infty ,-\\frac{8}{3}\\right]\\cup \\left[6,\\infty \\right)\\\\[\/latex]<\/span><\/p>\n

35.\u00a0[latex]\\left(-\\infty ,-\\frac{8}{3}\\right]\\cup \\left[16,\\infty \\right)\\\\[\/latex]<\/span><\/p>\n37.\n\"Graph\n\n39.\n\"Graph\n\n41.\n\"Graph\n\n43.\n\"Graph\n\n45.\n\"Graph\n\n47.\n\"Graph\n\n49.\n\"Graph\n\n51.\n\"Graph\n\n53.\u00a0range: [latex]\\left[0,20\\right][\/latex]\n\"Graph\n\n55.\u00a0[latex]x\\text{-}[\/latex] intercepts:\n\"Graph\n\n57.\u00a0[latex]\\left(-\\infty ,\\infty \\right)[\/latex]\n\n59.\u00a0There 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].\n\n61.\u00a0[latex]|p - 0.08|\\le 0.015[\/latex]\n\n63.\u00a0[latex]|x - 5.0|\\le 0.01[\/latex]","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0[latex]|x - 2|\\le 3\\\\[\/latex]<\/span><\/p>\n

2.\u00a0using the variable [latex]p\\\\[\/latex] for passing, [latex]|p - 80|\\le 20\\\\[\/latex]<\/span><\/p>\n

3.\u00a0[latex]f\\left(x\\right)=-|x+2|+3\\\\[\/latex]<\/span><\/p>\n

4.\u00a0[latex]x=-1\\\\[\/latex] or [latex]x=2\\\\[\/latex]<\/span><\/p>\n

5.\u00a0[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].<\/span><\/p>\n

6.\u00a0[latex]4\\le x\\le 8\\\\[\/latex]<\/span><\/p>\n

7.\u00a0[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]<\/span><\/p>\n

Solutions to Odd-Numbered Exercises<\/b><\/span><\/p>\n

1.\u00a0Isolate 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.<\/span><\/p>\n

3.\u00a0The 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.<\/span><\/p>\n

5.\u00a0First 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.<\/span><\/p>\n

7.\u00a0[latex]|x+4|=\\frac{1}{2}\\\\[\/latex]<\/span><\/p>\n

9.\u00a0[latex]|f\\left(x\\right)-8|<0.03\\\\[\/latex]<\/span><\/p>\n

11.\u00a0[latex]\\left\\{1,11\\right\\}\\\\[\/latex]<\/span><\/p>\n

13.\u00a0[latex]\\left\\{\\frac{9}{4},\\frac{13}{4}\\right\\}\\\\[\/latex]<\/span><\/p>\n

15.\u00a0[latex]\\left\\{\\frac{10}{3},\\frac{20}{3}\\right\\}\\\\[\/latex]<\/span><\/p>\n

17.\u00a0[latex]\\left\\{\\frac{11}{5},\\frac{29}{5}\\right\\}\\\\[\/latex]<\/span><\/p>\n

19.\u00a0[latex]\\left\\{\\frac{5}{2},\\frac{7}{2}\\right\\}\\\\[\/latex]<\/span><\/p>\n

21.\u00a0No solution<\/span><\/p>\n

23.\u00a0[latex]\\left\\{-57,27\\right\\}\\\\[\/latex]<\/span><\/p>\n

25.\u00a0[latex]\\left(0,-8\\right);\\left(-6,0\\right),\\left(4,0\\right)\\\\[\/latex]<\/span><\/p>\n

27.\u00a0[latex]\\left(0,-7\\right)\\\\[\/latex]; no [latex]x\\\\[\/latex] -intercepts<\/span><\/p>\n

29.\u00a0[latex]\\left(-\\infty ,-8\\right)\\cup \\left(12,\\infty \\right)\\\\[\/latex]<\/span><\/p>\n

31.\u00a0[latex]\\frac{-4}{3}\\le x\\le 4\\\\[\/latex]<\/span><\/p>\n

33.\u00a0[latex]\\left(-\\infty ,-\\frac{8}{3}\\right]\\cup \\left[6,\\infty \\right)\\\\[\/latex]<\/span><\/p>\n

35.\u00a0[latex]\\left(-\\infty ,-\\frac{8}{3}\\right]\\cup \\left[16,\\infty \\right)\\\\[\/latex]<\/span><\/p>\n

37.
\n\"Graph<\/p>\n

39.
\n\"Graph<\/p>\n

41.
\n\"Graph<\/p>\n

43.
\n\"Graph<\/p>\n

45.
\n\"Graph<\/p>\n

47.
\n\"Graph<\/p>\n

49.
\n\"Graph<\/p>\n

51.
\n\"Graph<\/p>\n

53.\u00a0range: [latex]\\left[0,20\\right][\/latex]
\n\"Graph<\/p>\n

55.\u00a0[latex]x\\text{-}[\/latex] intercepts:
\n\"Graph<\/p>\n

57.\u00a0[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/p>\n

59.\u00a0There 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].<\/p>\n

61.\u00a0[latex]|p - 0.08|\\le 0.015[\/latex]<\/p>\n

63.\u00a0[latex]|x - 5.0|\\le 0.01[\/latex]<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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