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Difficulty: MediumLinear Inequalities and Absolute Value

If xx is a real number that satisfies the inequality ∣5−2x∣≤7|5 - 2x| \le 7, what is the minimum possible value of the expression 3−4x3 - 4x?

  1. −21-21Answer
  2. B
    −1-1
  3. C
    77
  4. D
    2727
  5. E
    −27-27

Answer

−21-21
Solving ∣5−2x∣≤7|5 - 2x| \le 7 yields −7≤5−2x≤7-7 \le 5 - 2x \le 7. Subtracting 55 gives −12≤−2x≤2-12 \le -2x \le 2. Dividing by −2-2 and flipping the inequality direction gives −1≤x≤6-1 \le x \le 6. To minimize 3−4x3 - 4x, we choose the maximum value of xx because the term −4x-4x decreases as xx increases. Substituting x=6x = 6 yields 3−4(6)=−213 - 4(6) = -21.

Step-by-Step Solution

1
Unpack the absolute value inequality into a compound inequality
−7≤5−2x≤7-7 \le 5 - 2x \le 7
For any real constant k≥0k \ge 0, ∣u∣≤k|u| \le k is equivalent to −k≤u≤k-k \le u \le k.
2
Isolate the variable term by subtracting 55 from all parts of the inequality
−12≤−2x≤2-12 \le -2x \le 2
Subtracting a constant from all parts preserves inequality direction.
3
Divide all parts by −2-2 and reverse the inequality signs
6≥x≥−16 \ge x \ge -1, which is equivalent to −1≤x≤6-1 \le x \le 6
Dividing an inequality by a negative number reverses the direction of the inequality signs.
4
Find the minimum value of 3−4x3 - 4x over the interval [−1,6][-1, 6]
The minimum value occurs at x=6x = 6: 3−4(6)=3−24=−213 - 4(6) = 3 - 24 = -21
Because the linear expression 3−4x3 - 4x has a negative coefficient for xx, it is a decreasing function; its minimum occurs at the largest allowed value of xx.

Key Concept

Linear Inequalities and Absolute Value Bounds
Estimated Time:1m 30s
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Topics

Linear Inequalities and Absolute Value

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