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Difficulty: MediumAbsolute Value Equations and Inequalities

Which of the following inequality expressions represents the complete solution set for pp in the inequality 15−3∣2p+5∣<615 - 3|2p + 5| < 6?

  1. A
    p>−1p > -1
  2. B
    −4<p<−1-4 < p < -1
  3. p<−4p < -4 or p>−1p > -1Answer
  4. D
    −114<p<−94-\frac{11}{4} < p < -\frac{9}{4}
  5. E
    p<−1p < -1

Answer

p<−4p < -4 or p>−1p > -1
The correct solution is obtained by first isolating the absolute value expression. Subtracting 15 from both sides of the inequality 15−3∣2p+5∣<615 - 3|2p + 5| < 6 yields −3∣2p+5∣<−9-3|2p + 5| < -9. Dividing both sides by −3-3 and reversing the inequality sign gives ∣2p+5∣>3|2p + 5| > 3. This absolute value inequality splits into two cases: 2p+5>32p + 5 > 3 (which simplifies to p>−1p > -1) or 2p+5<−32p + 5 < -3 (which simplifies to p<−4p < -4). Combining these yields the complete solution set p<−4p < -4 or p>−1p > -1.

Step-by-Step Solution

1
Subtract 15 from both sides of the inequality 15−3∣2p+5∣<615 - 3|2p + 5| < 6.
−3∣2p+5∣<−9-3|2p + 5| < -9
To isolate the absolute value term, first subtract the constant term from both sides.
2
Divide both sides of −3∣2p+5∣<−9-3|2p + 5| < -9 by −3-3 and reverse the inequality sign.
∣2p+5∣>3|2p + 5| > 3
Dividing an inequality by a negative number requires reversing the direction of the inequality sign.
3
Solve the absolute value inequality ∣2p+5∣>3|2p + 5| > 3 by setting up two separate inequalities.
2p+5>32p + 5 > 3 or 2p+5<−32p + 5 < -3
An absolute value inequality of the form ∣u∣>c|u| > c splits into u>cu > c or u<−cu < -c.
4
Solve each linear inequality for pp.
p>−1p > -1 or p<−4p < -4
Subtract 5 from both sides and then divide by 2 for both inequalities to isolate pp.

Key Concept

Solving absolute value inequalities involving algebraic manipulation and reversing the inequality sign when multiplying or dividing by a negative number.
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Topics

Absolute Value Equations and Inequalities

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