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Difficulty: MediumLinear Inequalities and Absolute Value

If xx is a real number that satisfies the inequality −2∣3x−4∣+5≥−7-2|3x - 4| + 5 \ge -7, which of the following inequalities represents all possible values of xx?

  1. −23≤x≤103-\frac{2}{3} \le x \le \frac{10}{3}Answer
  2. B
    x≤−23x \le -\frac{2}{3} or x≥103x \ge \frac{10}{3}
  3. C
    −103≤x≤23-\frac{10}{3} \le x \le \frac{2}{3}
  4. D
    x≤103x \le \frac{10}{3}
  5. E
    −13≤x≤3-\frac{1}{3} \le x \le 3

Answer

−23≤x≤103-\frac{2}{3} \le x \le \frac{10}{3}
Isolating ∣3x−4∣|3x - 4| requires dividing −2∣3x−4∣≥−12-2|3x - 4| \ge -12 by −2-2, which reverses the inequality to ∣3x−4∣≤6|3x - 4| \le 6. Expanding this into the compound inequality −6≤3x−4≤6-6 \le 3x - 4 \le 6 and solving for xx yields the interval −23≤x≤103-\frac{2}{3} \le x \le \frac{10}{3}.

Step-by-Step Solution

1
Isolate the absolute value expression by subtracting 5 from both sides of the inequality.
−2∣3x−4∣≥−12-2|3x - 4| \ge -12
Before removing absolute value bars, the term ∣3x−4∣|3x - 4| must be isolated on one side.
2
Divide both sides by −2-2 and reverse the inequality sign.
∣3x−4∣≤6|3x - 4| \le 6
Dividing an inequality by a negative number reverses the direction of the inequality sign.
3
Express the absolute value inequality ∣u∣≤k|u| \le k as a double inequality −k≤u≤k-k \le u \le k.
−6≤3x−4≤6-6 \le 3x - 4 \le 6
The distance of 3x−43x - 4 from 0 on the number line must be at most 6 units.
4
Add 4 to all three parts of the inequality and divide by 3.
−2≤3x≤10  ⟹  −23≤x≤103-2 \le 3x \le 10 \implies -\frac{2}{3} \le x \le \frac{10}{3}
Solving for xx isolates the variable in the center of the double inequality.

Key Concept

Solving linear absolute value inequalities requires reversing the inequality sign when multiplying or dividing by negative numbers, and expressing ∣A∣≤B|A| \le B as −B≤A≤B-B \le A \le B.
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Topics

Linear Inequalities and Absolute Value

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