# 1.7e: Exercises - Absolute Value

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### A: Absolute Value Equations (I)

Exercise $$\PageIndex{A}$$

$$\bigstar$$ Solve the following absolute value equations.

 $$|x−5| = 8$$ $$|x−2| = 4$$ $$|x+4| = 3$$ $$|x+2| = 11$$ $$|x+2|−3 = 4$$ $$|4−x|+5 = 12$$ $$2|x−7|+5=9$$ $$3|x+5| = 6$$ $$3|x−4|−4=8$$ $$4|x−1|+2=10$$ $$3|x−4|+2=11$$ $$3|4x−5|−4=11$$ $$3|x+2|−5=4$$ $$|2x−3|−4=1$$ $$|3x−5|−1=6$$ $$|5x−4|−3=8$$ $$|4x−3|−5=2$$ $$−2|x−3|+8=−4$$ $$−2|3−2x| = −6$$ $$|3x−4|+5=7$$ $$|4x+7|+2=5$$ $$|34x−3|+7=2$$ $$|35x−2|+5=2$$ $$|12x+5|+4=1$$ $$|4x−1|−3=0$$ $$|14x+3|+3=1$$
 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.

 $$−3 \left| \dfrac{x}{2}−4 \right|+4=−5$$ $$\left| \dfrac{2}{3}x−4\right|-11=3$$. $$\left| \dfrac{x}{3}−\dfrac{1}{4}\right| = \dfrac{1}{12}$$ $$\left| \dfrac{x}{4}−\dfrac{1}{2}\right|= \dfrac{2}{3}$$ $$\left| \dfrac{3}{4}x−5\right|- 9=4$$ $$\left| \dfrac{5}{6}x+6 \right|=8$$  $$|x+2| = \dfrac{1}{3}x+5$$ $$|x−3|=5−\dfrac{1}{2}x$$ $$|x−2| = \dfrac{1}{3}x+2$$ $$|x+4| = \dfrac{1}{3}x+4$$ $$|4x+3|=|2x+1|$$ $$|3x−2| = |2x−3|$$ $$|6−x|=|3−2x|$$ $$|6x−5|=|2x+3|$$ $$|5x−1|=|2x+3|$$ $$|7x−3|=|3x+7|$$ $$|6x−5|=|3x+4|$$ $$3|x+2|−5 = |x+2|+7$$ $$4−3|4−x| = 2|4−x|−1$$
 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.

 $$|x| < 5$$ $$|x| ≤ 2$$ $$|x + 3| ≤ 1$$ $$|x − 7| < 8$$ $$|x − 5| < 0$$ $$|x + 8| < −7$$ $$|2x − 3| ≤ 5$$ $$|3x − 9| < 27$$ $$|5x − 3| ≤ 0$$ $$|10x + 5| < 25$$ $$\left| \dfrac { 1 } { 3 } x - \dfrac { 2 } { 3 } \right| \leq 1 \\[4pt]$$ $$\left| \dfrac { 1 } { 12 } x - \dfrac { 1 } { 2 } \right| \leq \dfrac { 3 } { 2 }$$ $$|x| ≥ 5$$ $$|x| > 1$$ $$|x + 2| > 8$$ $$|x − 7| ≥ 11$$ $$|x + 5| ≥ 0$$ $$|x − 12| > −4$$ $$|2x − 5| ≥ 9$$ $$|2x + 3| ≥ 15$$ $$|4x − 3| > 9$$ $$|3x − 7| ≥ 2$$ $$\left| \dfrac { 1 } { 7 } x - \dfrac { 3 } { 14 } \right| > \dfrac { 1 } { 2 } \\[4pt]$$ $$\left| \dfrac { 1 } { 2 } x + \dfrac { 5 } { 4 } \right| > \dfrac { 3 } { 4 }$$

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.

 $$|3 (2x − 1)| > 15$$ $$|3 (x − 3)| ≤ 21$$ $$−5 |x − 4| > −15$$ $$−3 |x + 8| ≤ −18$$ $$6 − 3 |x − 4| < 3$$ $$5 − 2 |x + 4| ≤ −7$$ $$1+ |2x + 5| > 12$$ $$2 + |3x − 7| ≤ 9$$ $$|2x + 25| − 4 ≥ 9$$ $$|3 (x − 3)| − 8 < −2$$ $$2 |9x + 5| + 8 > 6$$ $$3 |4x − 9| + 4 < −1$$ $$5 |4 − 3x| − 10 ≤ 0$$ $$6 |1 − 4x| − 24 ≥ 0$$ $$3 − 2 |x + 7| > −7$$ $$9 − 7 |x − 4| < −12$$ $$|5 (x − 4) + 5| > 15$$ $$|3 (x − 9) + 6| ≤ 3$$ $$7 − |−4 + 2 (3 − 4x)| > 5$$ $$9 − |6 + 3 (2x − 1)| ≥ 8$$ $$12 + 4 |2x − 1| ≤ 12$$ $$3 − 6 |3x − 2| ≥ 3$$ $$\dfrac{1}{2} |2x − 1| + 3 < 4$$ $$2 \left| \dfrac{1}{2} x + \dfrac{2}{3} \right| − 3 ≤ −1$$ $$\left| \dfrac { 1 } { 3 } ( x + 2 ) - \dfrac { 7 } { 6 } \right| - \dfrac { 2 } { 3 } \leq - \dfrac { 1 } { 6 }$$ $$\left| \dfrac { 1 } { 10 } ( x + 3 ) - \dfrac { 1 } { 2 } \right| + \dfrac { 3 } { 20 } > \dfrac { 1 } { 4 }$$ $$\dfrac { 3 } { 2 } - \left| 2 - \dfrac { 1 } { 3 } x \right| < \dfrac { 1 } { 2 }$$ $$\dfrac { 5 } { 4 } - \left| \dfrac { 1 } { 2 } - \dfrac { 1 } { 4 } x \right| < \dfrac { 3 } { 8 }$$

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

83. $$( 1,7 )$$;

85. $$( - \infty , 3 ) \cup ( 5 , \infty )$$;

87. $$( - \infty , - 8 ) \cup ( 3 , \infty )$$;

89. $$( - \infty , - 19 ] \cup [ - 6 , \infty )$$;

91. $$\mathbb { R }$$;

93. $$\left[ \frac { 2 } { 3 } , 2 \right]$$;

95. $$( - 12 , - 2 )$$;

97. $$( - \infty , 0 ) \cup ( 6 , \infty )$$;

99.  $$\left( 0 , \frac { 1 } { 2 } \right)$$;

101. $$\frac { 1 } { 2 }$$;

103. $$\left( - \frac { 1 } { 2 } , \frac { 3 } { 2 } \right)$$;

105. $$[ 0,3 ]$$;

107. $$( - \infty , 3 ) \cup ( 9 , \infty )$$;

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