1.7e: Exercises - Absolute Value

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A: Absolute Value Equations (I)

Exercise \(\PageIndex{A}\)

\( \bigstar \) Solve the following absolute value equations.

\(|x−5| = 8\)
\(|x−2| = 4\)
\(|x+4| = 3\)
\(|x+2| = 11\)
\(|x+2|−3 = 4 \)
\(|4−x|+5 = 12\)

\(2|x−7|+5=9\)
\(3|x+5| = 6\)
\(3|x−4|−4=8\)
\(4|x−1|+2=10\)
\(3|x−4|+2=11\)
\(3|4x−5|−4=11\)

\(3|x+2|−5=4\)
\(|2x−3|−4=1\)
\(|3x−5|−1=6\)
\(|5x−4|−3=8\)
\(|4x−3|−5=2\)
\(−2|x−3|+8=−4\)
\(−2|3−2x| = −6\)

\(|3x−4|+5=7\)
\(|4x+7|+2=5\)
\(|34x−3|+7=2\)
\(|35x−2|+5=2\)
\(|12x+5|+4=1\)
\(|4x−1|−3=0\)
\(|14x+3|+3=1\)

Answers to odd exercises:

1. \(x = −3\) or \(x = 13\)
3. \(x = −7 \) or \( x = −1\)
5. \(x = −9 \) or \( x = 5\)
7. \(x=5\) or \(x=9\)

9. \(x=8,\space x=0\)
11. \(x=7, \, x=1\)
13. \(x=1, \,x=−5\)

15. \(x=4, \space x=−\dfrac{2}{3}\)
17. \(x=−1,\space x=\dfrac{5}{2}\)
19. \(x = 0\) or \(x = 3\)

21. \(x=−1, \,x=−\dfrac{5}{2}\)
23. no solution
25. \(x=1, \,x=−\dfrac{1}{2}\)

\( \bigstar \) Solve the following absolute value equations.

\(−3 \left| \dfrac{x}{2}−4 \right|+4=−5\)
\( \left| \dfrac{2}{3}x−4\right|-11=3\).
\( \left| \dfrac{x}{3}−\dfrac{1}{4}\right| = \dfrac{1}{12}\)
\( \left| \dfrac{x}{4}−\dfrac{1}{2}\right|= \dfrac{2}{3}\)

\( \left| \dfrac{3}{4}x−5\right|- 9=4\)
\( \left| \dfrac{5}{6}x+6 \right|=8\)
\(|x+2| = \dfrac{1}{3}x+5\)
\(|x−3|=5−\dfrac{1}{2}x\)

\(|x−2| = \dfrac{1}{3}x+2\)
\(|x+4| = \dfrac{1}{3}x+4\)
\(|4x+3|=|2x+1|\)
\(|3x−2| = |2x−3|\)
\(|6−x|=|3−2x|\)

\(|6x−5|=|2x+3|\)
\(|5x−1|=|2x+3|\)
\(|7x−3|=|3x+7|\)
\(|6x−5|=|3x+4|\)
\(3|x+2|−5 = |x+2|+7\)
\(4−3|4−x| = 2|4−x|−1\)

Answers to odd exercises:

31. \(x=14, \,x=2\)
33. \(x = \frac{1}{2}, \; x = 1\)

35. \(x = \frac{-32}{3}, \; x = 24\)
37. \(x = \frac{9}{2}, \; x = \frac{-21}{4} \)

39. \(x = 0\), \(x = 6\)
41. \(x=−1, \,x=−\frac{2}{3}\)

43. \(x=−3, \,x=3\)
45. \(x=−\frac{2}{7}, \; x=\frac{4}{3}\)

47. \(x=3, x=\frac{1}{9}\)
49. \(x = 3, \; x = 5\)

B: Absolute Value Linear Inequalities (I)

Exercise \(\PageIndex{B}\): Absolute Value Linear Inequalities I

\( \bigstar \) Solve. State the solution in interval notation and graph the solution set on the number line.

\(|x| < 5\)
\(|x| ≤ 2\)
\(|x + 3| ≤ 1\)
\(|x − 7| < 8\)
\(|x − 5| < 0\)
\(|x + 8| < −7\)
\(|2x − 3| ≤ 5\)

\(|3x − 9| < 27\)
\(|5x − 3| ≤ 0\)
\(|10x + 5| < 25 \)
\(\left| \dfrac { 1 } { 3 } x - \dfrac { 2 } { 3 } \right| \leq 1 \\[4pt]\)
\(\left| \dfrac { 1 } { 12 } x - \dfrac { 1 } { 2 } \right| \leq \dfrac { 3 } { 2 }\)

\(|x| ≥ 5\)
\(|x| > 1\)
\(|x + 2| > 8\)
\(|x − 7| ≥ 11\)
\(|x + 5| ≥ 0\)
\(|x − 12| > −4\)
\(|2x − 5| ≥ 9\)

\(|2x + 3| ≥ 15\)
\(|4x − 3| > 9 \)
\(|3x − 7| ≥ 2 \)
\(\left| \dfrac { 1 } { 7 } x - \dfrac { 3 } { 14 } \right| > \dfrac { 1 } { 2 } \\[4pt] \)
\(\left| \dfrac { 1 } { 2 } x + \dfrac { 5 } { 4 } \right| > \dfrac { 3 } { 4 }\)

Answers to odd exercises:

51. \(( - 5,5 )\);

53. \([ - 4 , - 2 ]\);

55. \(\emptyset\);

57. \([ - 1,4 ]\);

59. \(\left\{ \frac { 3 } { 5 } \right\}\);

61. \([ - 1,5 ]\);

63. \(( - \infty , - 5 ] \cup [ 5 , \infty )\);

65. \(( - \infty , - 10 ) \cup ( 6 , \infty )\);

67. \(\mathbb { R }\);

69. \(( - \infty , - 2 ] \cup [ 7 , \infty )\);

71. \(\left( - \infty , - \frac { 3 } { 2 } \right) \cup ( 3 , \infty )\);

73. \(( - \infty , - 2 ) \cup ( 5 , \infty )\);

C: Absolute Value Linear Inequalities (II)

Exercise \(\PageIndex{C}\): Absolute Value Linear Inequalities II

\( \bigstar \) Solve. State the solution in interval notation and graph the solution set on the number line.

\(|3 (2x − 1)| > 15\)
\(|3 (x − 3)| ≤ 21\)
\(−5 |x − 4| > −15\)
\(−3 |x + 8| ≤ −18\)
\(6 − 3 |x − 4| < 3\)
\(5 − 2 |x + 4| ≤ −7\)
\(1+ |2x + 5| > 12\)
\(2 + |3x − 7| ≤ 9\)

\(|2x + 25| − 4 ≥ 9\)
\(|3 (x − 3)| − 8 < −2\)
\(2 |9x + 5| + 8 > 6\)
\(3 |4x − 9| + 4 < −1\)
\(5 |4 − 3x| − 10 ≤ 0\)
\(6 |1 − 4x| − 24 ≥ 0\)
\(3 − 2 |x + 7| > −7\)
\(9 − 7 |x − 4| < −12\)

\(|5 (x − 4) + 5| > 15\)
\(|3 (x − 9) + 6| ≤ 3\)
\(7 − |−4 + 2 (3 − 4x)| > 5\)
\(9 − |6 + 3 (2x − 1)| ≥ 8\)
\(12 + 4 |2x − 1| ≤ 12\)
\(3 − 6 |3x − 2| ≥ 3\)
\(\dfrac{1}{2} |2x − 1| + 3 < 4\)

\(2 \left| \dfrac{1}{2} x + \dfrac{2}{3} \right| − 3 ≤ −1\)
\(\left| \dfrac { 1 } { 3 } ( x + 2 ) - \dfrac { 7 } { 6 } \right| - \dfrac { 2 } { 3 } \leq - \dfrac { 1 } { 6 }\)
\(\left| \dfrac { 1 } { 10 } ( x + 3 ) - \dfrac { 1 } { 2 } \right| + \dfrac { 3 } { 20 } > \dfrac { 1 } { 4 }\)
\(\dfrac { 3 } { 2 } - \left| 2 - \dfrac { 1 } { 3 } x \right| < \dfrac { 1 } { 2 }\)
\(\dfrac { 5 } { 4 } - \left| \dfrac { 1 } { 2 } - \dfrac { 1 } { 4 } x \right| < \dfrac { 3 } { 8 }\)

Answers to odd exercises:

81. \(( - \infty , - 2 ) \cup ( 3 , \infty )\);