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Difficulty: HardLinear Inequalities and Absolute Value

If xx is an integer that satisfies both ∣2x−9∣≤15|2x - 9| \le 15 and 5−2x−3≤3\frac{5 - 2x}{-3} \le 3, what is the product of the smallest and largest possible values of xx?

  1. −21-21Answer
  2. B
    8484
  3. C
    3535
  4. D
    −36-36
  5. E
    2121

Answer

The product of the smallest and largest possible integer values of xx is −21-21.
Solving ∣2x−9∣≤15|2x - 9| \le 15 gives −3≤x≤12-3 \le x \le 12. Solving 5−2x−3≤3\frac{5 - 2x}{-3} \le 3 requires reversing the inequality sign twice (first when multiplying by −3-3, then when dividing by −2-2), which yields x≤7x \le 7. Taking the intersection of both intervals gives −3≤x≤7-3 \le x \le 7. The smallest integer in this range is −3-3 and the largest is 77, giving a product of (−3)×7=−21(-3) \times 7 = -21.

Step-by-Step Solution

1
Solve the absolute value inequality ∣2x−9∣≤15|2x - 9| \le 15.
−15≤2x−9≤15  ⟹  −6≤2x≤24  ⟹  −3≤x≤12-15 \le 2x - 9 \le 15 \implies -6 \le 2x \le 24 \implies -3 \le x \le 12.
An absolute value inequality ∣u∣≤a|u| \le a (for a≥0a \ge 0) is equivalent to the compound inequality −a≤u≤a-a \le u \le a.
2
Solve the linear inequality 5−2x−3≤3\frac{5 - 2x}{-3} \le 3.
Multiply by −3-3 and flip the inequality direction: 5−2x≥−95 - 2x \ge -9. Subtract 55: −2x≥−14-2x \ge -14. Divide by −2-2 and flip the inequality direction again: x≤7x \le 7.
Multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign.
3
Determine the intersection of the two solution sets.
Combining −3≤x≤12-3 \le x \le 12 and x≤7x \le 7 gives −3≤x≤7-3 \le x \le 7.
The value of xx must satisfy both conditions simultaneously.
4
Identify the smallest and largest integer values of xx and calculate their product.
Smallest integer =−3= -3, largest integer =7= 7. Product =(−3)×7=−21= (-3) \times 7 = -21.
Both endpoints −3-3 and 77 are included in the closed interval [−3,7][-3, 7].

Key Concept

Solving compound linear inequalities involving absolute values and applying the rule for reversing inequality signs when multiplying or dividing by negative numbers.
Estimated Time:2m 0s
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Topics

Linear Inequalities and Absolute Value

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