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Difficulty: EasyAbsolute Value Equations and Inequalities

Which of the following inequality expressions represents the complete set of real values of pp that satisfy the inequality ∣7−2p∣<13|7 - 2p| < 13?

  1. A
    p<−3p < -3 or p>10p > 10
  2. B
    p>−3p > -3
  3. −3<p<10-3 < p < 10Answer
  4. D
    −135<p<135-\frac{13}{5} < p < \frac{13}{5}
  5. E
    −10<p<3-10 < p < 3

Answer

−3<p<10-3 < p < 10
To solve ∣7−2p∣<13|7 - 2p| < 13, write it as the compound inequality −13<7−2p<13-13 < 7 - 2p < 13. Subtracting 77 from all parts gives −20<−2p<6-20 < -2p < 6. Dividing by −2-2 and reversing the inequality signs yields −3<p<10-3 < p < 10.

Step-by-Step Solution

1
Write the absolute value inequality as a compound inequality.
−13<7−2p<13-13 < 7 - 2p < 13
An absolute value inequality of the form ∣x∣<c|x| < c is equivalent to −c<x<c-c < x < c.
2
Subtract 77 from all three parts of the compound inequality.
−20<−2p<6-20 < -2p < 6
To isolate the variable term, we perform the inverse operation of adding 77 by subtracting 77 from each part.
3
Divide all parts by −2-2 and reverse the inequality signs.
10>p>−310 > p > -3, which simplifies to −3<p<10-3 < p < 10
Dividing an inequality by a negative number requires reversing the direction of the inequality signs to preserve the truth of the statement.

Key Concept

Solving absolute value inequalities of the form ∣ax+b∣<c|ax + b| < c
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Topics

Absolute Value Equations and Inequalities

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