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Difficulty: HardAbsolute Value Equations and Inequalities

For all real values of tt that satisfy the inequality 12−4∣3−2t∣≤−812 - 4|3 - 2t| \leq -8, which of the following inequalities represents the complete set of possible values of tt?

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

Answer

t≤−1t \leq -1 or t≥4t \geq 4
The correct answer is t≤−1t \leq -1 or t≥4t \geq 4. To solve the inequality, we first subtract 12 from both sides to get −4∣3−2t∣≤−20-4|3 - 2t| \leq -20. Next, dividing both sides by −4-4 and reversing the inequality sign gives ∣3−2t∣≥5|3 - 2t| \geq 5. This absolute value inequality splits into two cases: 3−2t≥53 - 2t \geq 5 or 3−2t≤−53 - 2t \leq -5. Solving the first case, we subtract 3 to get −2t≥2-2t \geq 2, and dividing by −2-2 while reversing the inequality sign yields t≤−1t \leq -1. Solving the second case, we subtract 3 to get −2t≤−8-2t \leq -8, and dividing by −2-2 while reversing the inequality sign yields t≥4t \geq 4. Combining these, we obtain the solution set t≤−1t \leq -1 or t≥4t \geq 4.

Step-by-Step Solution

1
Isolate the absolute value term by subtracting 12 from both sides of the inequality.
−4∣3−2t∣≤−20-4|3 - 2t| \leq -20
Before splitting an absolute value inequality, the absolute value expression must be isolated on one side.
2
Divide both sides of the inequality by −4-4 and reverse the inequality sign.
∣3−2t∣≥5|3 - 2t| \geq 5
Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign.
3
Split the absolute value inequality ∣u∣≥c|u| \geq c (where c>0c > 0) into two separate inequalities: u≥cu \geq c or u≤−cu \leq -c.
3−2t≥53 - 2t \geq 5 or 3−2t≤−53 - 2t \leq -5
An absolute value inequality of the form ∣u∣≥c|u| \geq c represents values that are at least cc units away from zero, which lie in two disjoint intervals.
4
Solve the first inequality: 3−2t≥53 - 2t \geq 5.
−2t≥2  ⟹  t≤−1-2t \geq 2 \implies t \leq -1
Subtracting 3 from both sides gives −2t≥2-2t \geq 2. Dividing by −2-2 and reversing the inequality sign yields t≤−1t \leq -1.
5
Solve the second inequality: 3−2t≤−53 - 2t \leq -5.
−2t≤−8  ⟹  t≥4-2t \leq -8 \implies t \geq 4
Subtracting 3 from both sides gives −2t≤−8-2t \leq -8. Dividing by −2-2 and reversing the inequality sign yields t≥4t \geq 4.
6
Combine the solutions from both cases to express the complete solution set.
t≤−1t \leq -1 or t≥4t \geq 4
The complete solution set is the union of the solutions to both cases.

Key Concept

Solving absolute value inequalities of the form ∣ax+b∣≥c|ax + b| \geq c by translating them into compound inequalities and carefully reversing the inequality sign when multiplying or dividing by negative numbers.
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Topics

Absolute Value Equations and Inequalities

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