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Difficulty: MediumLinear Inequalities and Absolute Value

If xx is a real number that satisfies the inequality ∣2x−7∣<5|2x - 7| < 5, which of the following represents all possible values of the expression 1−3x1 - 3x?

  1. −17<1−3x<−2-17 < 1 - 3x < -2Answer
  2. B
    −2<1−3x<17-2 < 1 - 3x < 17
  3. C
    −17<1−3x<1-17 < 1 - 3x < 1
  4. D
    −17<1−3x<17-17 < 1 - 3x < 17
  5. E
    −5<1−3x<5-5 < 1 - 3x < 5

Answer

−17<1−3x<−2-17 < 1 - 3x < -2
Solving ∣2x−7∣<5|2x - 7| < 5 gives −5<2x−7<5-5 < 2x - 7 < 5. Adding 77 to all three parts yields 2<2x<122 < 2x < 12, which simplifies to 1<x<61 < x < 6. Multiplying this inequality by −3-3 reverses the direction of the inequalities, resulting in −18<−3x<−3-18 < -3x < -3. Adding 11 to each part gives −17<1−3x<−2-17 < 1 - 3x < -2. Therefore, the range of possible values for the expression is strictly between −17-17 and −2-2.

Step-by-Step Solution

1
Express the absolute value inequality as a compound inequality.
−5<2x−7<5-5 < 2x - 7 < 5
An inequality of the form ∣u∣<k|u| < k for k>0k > 0 is equivalent to −k<u<k-k < u < k.
2
Isolate xx in the compound inequality.
1<x<61 < x < 6
Add 77 to all parts to get 2<2x<122 < 2x < 12, then divide all parts by 22 to obtain 1<x<61 < x < 6.
3
Multiply the compound inequality by −3-3.
−18<−3x<−3-18 < -3x < -3
Multiplying an inequality by a negative number reverses the direction of the inequality signs: −3(6)<−3(x)<−3(1)-3(6) < -3(x) < -3(1).
4
Add 11 to all parts of the compound inequality.
−17<1−3x<−2-17 < 1 - 3x < -2
Adding a constant to an inequality preserves the inequality direction.

Key Concept

Linear Inequalities and Absolute Value Transformations
Estimated Time:1m 30s
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Topics

Linear Inequalities and Absolute Value

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