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Difficulty: MediumAbsolute Value Equations and Inequalities

Which of the following is the set of all real values of pp for which the inequality ∣7−3p∣≥14|7 - 3p| \ge 14 is true?

  1. A
    −73≤p≤7-\frac{7}{3} \le p \le 7
  2. p≤−73 or p≥7p \le -\frac{7}{3} \text{ or } p \ge 7Answer
  3. C
    p≤−73p \le -\frac{7}{3}
  4. D
    p≤−72 or p≥72p \le -\frac{7}{2} \text{ or } p \ge \frac{7}{2}
  5. E
    p≥−73 or p≤7p \ge -\frac{7}{3} \text{ or } p \le 7

Answer

p≤−73 or p≥7p \le -\frac{7}{3} \text{ or } p \ge 7
To solve the inequality ∣7−3p∣≥14|7 - 3p| \ge 14, we split it into two separate inequalities: 7−3p≥147 - 3p \ge 14 or 7−3p≤−147 - 3p \le -14. Solving the first inequality gives −3p≥7-3p \ge 7, which simplifies to p≤−73p \le -\frac{7}{3} after dividing by −3-3 and reversing the inequality sign. Solving the second inequality gives −3p≤−21-3p \le -21, which simplifies to p≥7p \ge 7 after dividing by −3-3 and reversing the inequality sign. Combining these two cases yields the set of values p≤−73p \le -\frac{7}{3} or p≥7p \ge 7.

Step-by-Step Solution

1
Set up the two compound inequalities representing the absolute value inequality ∣7−3p∣≥14|7 - 3p| \ge 14.
7−3p≥147 - 3p \ge 14 or 7−3p≤−147 - 3p \le -14
An absolute value inequality of the form ∣u∣≥c|u| \ge c (where c>0c > 0) is equivalent to the union of u≥cu \ge c or u≤−cu \le -c.
2
Solve the first inequality 7−3p≥147 - 3p \ge 14.
p≤−73p \le -\frac{7}{3}
Subtract 77 from both sides to get −3p≥7-3p \ge 7. Then, divide by −3-3 and reverse the inequality sign because of division by a negative number.
3
Solve the second inequality 7−3p≤−147 - 3p \le -14.
p≥7p \ge 7
Subtract 77 from both sides to get −3p≤−21-3p \le -21. Then, divide by −3-3 and reverse the inequality sign because of division by a negative number.
4
Combine the two case solutions to state the final solution set.
p≤−73 or p≥7p \le -\frac{7}{3} \text{ or } p \ge 7
The complete solution set is the union of the solutions from both individual cases.

Key Concept

Solving absolute value inequalities of the form ∣ax+b∣≥c|ax + b| \ge c
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Topics

Absolute Value Equations and Inequalities

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