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Mathematics LibreTexts

1.7e: Exercises - Absolute Value

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    45456
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    A: Absolute Value Equations (I)

    Exercise \(\PageIndex{A}\)

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

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

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

    1. \(|x| < 5\)
    2. \(|x| ≤ 2\)
    3. \(|x + 3| ≤ 1\)
    4. \(|x − 7| < 8\)
    5. \(|x − 5| < 0\)
    6. \(|x + 8| < −7\)
    7. \(|2x − 3| ≤ 5\)
    1. \(|3x − 9| < 27\)
    2. \(|5x − 3| ≤ 0\)
    3. \(|10x + 5| < 25 \)
    4. \(\left| \dfrac { 1 } { 3 } x - \dfrac { 2 } { 3 } \right| \leq 1 \\[4pt]\)
    5. \(\left| \dfrac { 1 } { 12 } x - \dfrac { 1 } { 2 } \right| \leq \dfrac { 3 } { 2 }\)
    1. \(|x| ≥ 5\)
    2. \(|x| > 1\)
    3. \(|x + 2| > 8\)
    4. \(|x − 7| ≥ 11\)
    5. \(|x + 5| ≥ 0\)
    6. \(|x − 12| > −4\)
    7. \(|2x − 5| ≥ 9\)
    1. \(|2x + 3| ≥ 15\)
    2. \(|4x − 3| > 9 \)
    3. \(|3x − 7| ≥ 2 \)
    4. \(\left| \dfrac { 1 } { 7 } x - \dfrac { 3 } { 14 } \right| > \dfrac { 1 } { 2 } \\[4pt] \)
    5. \(\left| \dfrac { 1 } { 2 } x + \dfrac { 5 } { 4 } \right| > \dfrac { 3 } { 4 }\)
    Answers to odd exercises:

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

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

    55. \(\emptyset\);

    18dc3829d035ad7a33a362ec1a7e5f09.png

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

    4cb524885d5df3a2715ff992dcd5a6b3.png

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

    5fe987d3f662f3552c53cb4c800e585c.png

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

    e698c903acbeaae5e83c1bc9c6e8845a.png

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

    a7d47bfc737e19a60beecd34fdcbade7.png

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

    9d1fb16fd22556532e81ef67b14c73a1.png

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

    6d8a25b4d71237af99c7a5f08d9d21e2.png

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

    d1e745f298730b5b44f34dd56e033df2.png

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

    0010b88a922a03db183e099d487bc54b.png

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

    33b71142f5fcc934441a9b670c22e110.png
     

    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.

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

    Answers to odd exercises:

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

    ebcbc7e9a8c69eaf7dc93e0a895be02a.png

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

    6a84ac9f29153422122cecb7ae173a27.png

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

    291445ead16c4e9f28633d76d92ebd51.png

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

    22b7aed415f504f8381f18d4e7776f45.png

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

    13fa6f328decfd285cbe4425b06e4c93.png

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

    6d8a25b4d71237af99c7a5f08d9d21e2.png

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

    ceec08d41c2361f4613cd5e5d213844e.png

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

    9964b1ed11f681d97f2fa612af08d243.png

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

    755b3ba705f6729a468c1d53bd54200a.png

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

    629e476a686ff5681e41138257e2f57e.png

    101. \(\frac { 1 } { 2 }\);

    97abc5e7029c6c2abb111a8d6172fc3f.png

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

    9940bee4840619748881b6ee166fb60c.png

    105. \([ 0,3 ]\);

    1a67f6f2059288d7eb289b5b91888451.png

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

    6f9be38567ebfe5e3669e0e423e0f33b.png
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