Examkin

Question

Difficulty: HardAbsolute Value Equations and Inequalities

For all real values of ww that satisfy the inequality 4−∣3−2w∣≤−14 - |3 - 2w| \leq -1, which of the following expressions represents the complete set of possible values of ww?

  1. w≤−1w \leq -1 or w≥4w \geq 4Answer
  2. B
    −1≤w≤4-1 \leq w \leq 4
  3. C
    w≤−4w \leq -4 or w≥1w \geq 1
  4. D
    w≥4w \geq 4
  5. E
    w≤−1w \leq -1

Answer

The complete set of possible values is represented by the inequality w≤−1w \leq -1 or w≥4w \geq 4.
The correct answer is the solution set representing w≤−1w \leq -1 or w≥4w \geq 4. Isolating the absolute value expression yields ∣3−2w∣≥5|3 - 2w| \geq 5. This splits into two cases: 3−2w≥53 - 2w \geq 5 (which solves to w≤−1w \leq -1 after dividing by −2-2 and reversing the inequality sign) and 3−2w≤−53 - 2w \leq -5 (which solves to w≥4w \geq 4 after dividing by −2-2 and reversing the inequality sign). Combining these two cases gives the union w≤−1w \leq -1 or w≥4w \geq 4.

Step-by-Step Solution

1
Isolate the absolute value expression on one side of the inequality.
−∣3−2w∣≤−5-|3 - 2w| \leq -5, which simplifies to ∣3−2w∣≥5|3 - 2w| \geq 5 after multiplying by −1-1 and reversing the inequality sign.
Before splitting an absolute value inequality, the absolute value term must be isolated.
2
Split the absolute value inequality ∣3−2w∣≥5|3 - 2w| \geq 5 into two separate compound inequalities.
3−2w≥53 - 2w \geq 5 or 3−2w≤−53 - 2w \leq -5
An absolute value inequality of the form ∣u∣≥c|u| \geq c (where c>0c > 0) is equivalent to u≥cu \geq c or u≤−cu \leq -c.
3
Solve the first inequality: 3−2w≥53 - 2w \geq 5.
Subtracting 33 from both sides gives −2w≥2-2w \geq 2. Dividing both sides by −2-2 and reversing the inequality sign gives w≤−1w \leq -1.
Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign.
4
Solve the second inequality: 3−2w≤−53 - 2w \leq -5.
Subtracting 33 from both sides gives −2w≤−8-2w \leq -8. Dividing both sides by −2-2 and reversing the inequality sign gives w≥4w \geq 4.
Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign.
5
Combine the individual solutions to find the total solution set.
w≤−1w \leq -1 or w≥4w \geq 4
The solution to a 'greater than or equal to' absolute value inequality is the union of the solutions of the two split cases.

Key Concept

Solving absolute value inequalities with negative variable coefficients by isolating the absolute value term, splitting into cases, and reversing inequality signs when multiplying/dividing by a negative number.
Estimated Time:2m 0s
Rate this question

Topics

Absolute Value Equations and Inequalities

More questions from this topic