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Difficulty: Very hardLinear Inequalities and Absolute Value

If xx and yy are real numbers that satisfy the inequalities ∣3−2x∣≤9|3 - 2x| \le 9 and ∣4y+3∣≤13|4y + 3| \le 13, what is the maximum possible value of the expression ∣3x−4y∣|3x - 4y|?

Answer: 34

Answer

The maximum possible value of ∣3x−4y∣|3x - 4y| is 3434.
To find the maximum possible value of ∣3x−4y∣|3x - 4y|, we first solve for the ranges of xx and yy from their respective absolute value inequalities. From ∣3−2x∣≤9|3 - 2x| \le 9, we get −9≤3−2x≤9  ⟹  −3≤x≤6-9 \le 3 - 2x \le 9 \implies -3 \le x \le 6. From ∣4y+3∣≤13|4y + 3| \le 13, we get −13≤4y+3≤13  ⟹  −4≤y≤2.5-13 \le 4y + 3 \le 13 \implies -4 \le y \le 2.5. To maximize 3x−4y3x - 4y, we take the maximum value of 3x3x (3×6=183 \times 6 = 18) and the maximum value of −4y-4y (−4×−4=16-4 \times -4 = 16), giving 18+16=3418 + 16 = 34. To minimize 3x−4y3x - 4y, we take the minimum value of 3x3x (3×−3=−93 \times -3 = -9) and the minimum value of −4y-4y (−4×2.5=−10-4 \times 2.5 = -10), giving −9−10=−19-9 - 10 = -19. The expression 3x−4y3x - 4y ranges from −19-19 to 3434, so the maximum possible magnitude ∣3x−4y∣|3x - 4y| is max⁡(∣−19∣,∣34∣)=34\max(|-19|, |34|) = 34.

Step-by-Step Solution

1
Solve the inequality ∣3−2x∣≤9|3 - 2x| \le 9 for xx.
−3≤x≤6-3 \le x \le 6
Unpacking the absolute value yields −9≤3−2x≤9-9 \le 3 - 2x \le 9. Subtracting 33 gives −12≤−2x≤6-12 \le -2x \le 6. Dividing by −2-2 and reversing the inequality signs produces −3≤x≤6-3 \le x \le 6.
2
Solve the inequality ∣4y+3∣≤13|4y + 3| \le 13 for yy.
−4≤y≤2.5-4 \le y \le 2.5
Unpacking the absolute value yields −13≤4y+3≤13-13 \le 4y + 3 \le 13. Subtracting 33 gives −16≤4y≤10-16 \le 4y \le 10. Dividing by 44 gives −4≤y≤2.5-4 \le y \le 2.5.
3
Find the range of possible values for 3x3x and −4y-4y.
−9≤3x≤18-9 \le 3x \le 18 and −10≤−4y≤16-10 \le -4y \le 16
Multiplying −3≤x≤6-3 \le x \le 6 by 33 gives −9≤3x≤18-9 \le 3x \le 18. Multiplying −4≤y≤2.5-4 \le y \le 2.5 by −4-4 and flipping the signs gives −10≤−4y≤16-10 \le -4y \le 16.
4
Combine the bounds for 3x3x and −4y-4y to find the range for 3x−4y3x - 4y.
−19≤3x−4y≤34-19 \le 3x - 4y \le 34
The minimum value of 3x−4y3x - 4y is (−9)+(−10)=−19(-9) + (-10) = -19. The maximum value of 3x−4y3x - 4y is 18+16=3418 + 16 = 34.
5
Determine the maximum absolute value ∣3x−4y∣|3x - 4y| over the interval [−19,34][-19, 34].
34
The absolute value of any number in the interval [−19,34][-19, 34] ranges from 00 to max⁡(∣−19∣,∣34∣)=34\max(|-19|, |34|) = 34.

Key Concept

Absolute Value Inequalities and Expression Bounding
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Topics

Linear Inequalities and Absolute Value

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