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Difficulty: MediumAbsolute Value Equations and Inequalities

In the inequality 7−3∣2z+1∣≥−87 - 3|2z + 1| \ge -8, which of the following inequality expressions represents the complete solution set for zz?

  1. A
    z≤−3z \le -3 or z≥2z \ge 2
  2. B
    z≤2z \le 2
  3. −3≤z≤2-3 \le z \le 2Answer
  4. D
    −1.5≤z≤0.5-1.5 \le z \le 0.5
  5. E
    z≥2z \ge 2

Answer

−3≤z≤2-3 \le z \le 2
Subtracting 7 from both sides of 7−3∣2z+1∣≥−87 - 3|2z + 1| \ge -8 yields −3∣2z+1∣≥−15-3|2z + 1| \ge -15. Dividing both sides by −3-3 and reversing the inequality sign results in ∣2z+1∣≤5|2z + 1| \le 5. Writing this as the compound inequality −5≤2z+1≤5-5 \le 2z + 1 \le 5, then subtracting 1 and dividing by 2 yields the correct solution interval −3≤z≤2-3 \le z \le 2.

Step-by-Step Solution

1
Subtract 7 from both sides to begin isolating the absolute value term.
−3∣2z+1∣≥−15-3|2z + 1| \ge -15
To solve an absolute value inequality, we must first isolate the absolute value term on one side.
2
Divide both sides by -3 and reverse the direction of the inequality sign.
∣2z+1∣≤5|2z + 1| \le 5
Dividing an inequality by a negative number reverses the direction of the inequality sign.
3
Rewrite the absolute value inequality as a compound inequality.
−5≤2z+1≤5-5 \le 2z + 1 \le 5
An inequality of the form ∣u∣≤a|u| \le a (where a≥0a \ge 0) is equivalent to the compound inequality −a≤u≤a-a \le u \le a.
4
Subtract 1 from all three parts of the compound inequality.
−6≤2z≤4-6 \le 2z \le 4
This is the first step to isolate the variable zz in the middle.
5
Divide all three parts by 2.
−3≤z≤2-3 \le z \le 2
This fully isolates zz, giving the final solution interval.

Key Concept

Solving multi-step absolute value inequalities, including isolating the absolute value expression, reversing the inequality sign when dividing by a negative number, and expressing the solution as a compound inequality.
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Topics

Absolute Value Equations and Inequalities

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