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Difficulty: MediumAbsolute Value Equations and Inequalities

For all real values of qq that satisfy the inequality 8−2∣3q−1∣≥−68 - 2|3q - 1| \ge -6, which of the following inequality expressions represents the complete solution set for qq?

  1. $−2≤q≤83-2 \le q \le \frac{8}{3}Answer
  2. B
    q \le -2\text{ or }q \ge \frac{8}{3}
  3. C
    q≤83q \le \frac{8}{3}
  4. D
    q≥83q \ge \frac{8}{3}
  5. E
    q \ge 0

Answer

−2≤q≤83-2 \le q \le \frac{8}{3}
The correct answer is the interval containing all real numbers between −2-2 and 83\frac{8}{3} inclusive. This is determined by subtracting 8 from both sides of the inequality, dividing by −2-2 (which reverses the inequality sign to yield ∣3q−1∣≤7|3q - 1| \le 7), expressing this as the compound inequality −7≤3q−1≤7-7 \le 3q - 1 \le 7, and isolating qq.

Step-by-Step Solution

1
Subtract 8 from both sides of the inequality to isolate the absolute value term.
−2∣3q−1∣≥−14-2|3q - 1| \ge -14
To solve for the variable, we must first isolate the term containing the absolute value by performing inverse operations.
2
Divide both sides by −2-2 and reverse the inequality sign.
∣3q−1∣≤7|3q - 1| \le 7
Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.
3
Rewrite the absolute value inequality as a compound inequality.
−7≤3q−1≤7-7 \le 3q - 1 \le 7
An inequality of the form ∣x∣≤d|x| \le d (where d≥0d \ge 0) is equivalent to the compound inequality −d≤x≤d-d \le x \le d.
4
Add 1 to all three parts of the compound inequality.
−6≤3q≤8-6 \le 3q \le 8
Adding 1 eliminates the constant term from the middle section of the inequality.
5
Divide all three parts by 3 to solve for qq.
−2≤q≤83-2 \le q \le \frac{8}{3}
Dividing by 3 isolates the variable qq, yielding the complete solution set.

Key Concept

Solving multi-step absolute value inequalities, including reversing the inequality sign when dividing by a negative number and setting up a compound inequality to represent both positive and negative cases.
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Topics

Absolute Value Equations and Inequalities

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