Maximum Value of Modulus Expression

XAT 2023 · Quantitative Ability & Data Interpretation · Hard · Modulus Functions

Given $A = |x + 3| + |x - 2| - |2x - 8|$ . The maximum value of $|A|$ is:

Answer

Analyze critical points $x = -3, 2, 4$ .

$x \ge 4$ : $A = (x+3) + (x-2) - (2x-8) = 9$ .
$2 \le x < 4$ : $A = (x+3) + (x-2) + (2x-8) = 4x - 7 \in [1, 9)$ .
$-3 \le x < 2$ : $A = (x+3) - (x-2) + (2x-8) = 2x - 3 \in [-9, 1)$ .
$x < -3$ : $A = -(x+3) - (x-2) + (2x-8) = -9$ . In all cases, $|A| \le 9$ . Maximum value = 9.

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