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Difficulty: HardAbsolute Value Equations and Inequalities

What is the set of all real numbers yy that make the inequality 2−3∣1−2y∣≥−102 - 3|1 - 2y| \ge -10 a true statement?

  1. A
    −2.5≤y≤1.5-2.5 \le y \le 1.5
  2. B
    y≤−1.5y \le -1.5 or y≥2.5y \ge 2.5
  3. −1.5≤y≤2.5-1.5 \le y \le 2.5Answer
  4. D
    y≥−1.5y \ge -1.5
  5. E
    −4.5≤y≤5.5-4.5 \le y \le 5.5

Answer

−1.5≤y≤2.5-1.5 \le y \le 2.5
To solve the inequality 2−3∣1−2y∣≥−102 - 3|1 - 2y| \ge -10, we first isolate the absolute value term by subtracting 2 from both sides to get −3∣1−2y∣≥−12-3|1 - 2y| \ge -12, and then dividing both sides by −3-3. Since we divide by a negative number, the inequality sign reverses, giving ∣1−2y∣≤4|1 - 2y| \le 4. We rewrite this as the compound inequality −4≤1−2y≤4-4 \le 1 - 2y \le 4. Subtracting 1 from all parts yields −5≤−2y≤3-5 \le -2y \le 3. Finally, dividing by −2-2 and reversing the inequality signs gives 2.5≥y≥−1.52.5 \ge y \ge -1.5, which is rewritten from least to greatest as −1.5≤y≤2.5-1.5 \le y \le 2.5.

Step-by-Step Solution

1
Subtract 2 from both sides of the inequality to begin isolating the absolute value term.
−3∣1−2y∣≥−12-3|1 - 2y| \ge -12
Isolating the absolute value expression allows us to rewrite it as a standard inequality.
2
Divide both sides by −3-3 and reverse the inequality sign because of division by a negative number.
∣1−2y∣≤4|1 - 2y| \le 4
Dividing by a negative value requires reversing the inequality direction to preserve the truth of the statement.
3
Rewrite the absolute value inequality as a compound inequality.
−4≤1−2y≤4-4 \le 1 - 2y \le 4
An inequality of the form ∣u∣≤c|u| \le c (where c>0c > 0) is equivalent to the compound inequality −c≤u≤c-c \le u \le c.
4
Subtract 1 from all parts of the compound inequality.
−5≤−2y≤3-5 \le -2y \le 3
This is the next step to isolate the variable yy in the middle.
5
Divide all parts of the compound inequality by −2-2 and reverse the inequality signs.
2.5≥y≥−1.52.5 \ge y \ge -1.5, which is equivalent to −1.5≤y≤2.5-1.5 \le y \le 2.5
Dividing by the negative coefficient −2-2 requires reversing the direction of all inequality signs.

Key Concept

Solving absolute value inequalities involving negative coefficients by isolating the absolute value and reversing inequality signs when multiplying or dividing by negative numbers.
Estimated Time:1m 30s
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Topics

Absolute Value Equations and Inequalities

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