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Solve for x: |2x - 1| + |2x + 3| = 6.

1. x = -2 and x = 1
2. x = -1 and x = 2
3. x = -3/2 and x = 1/2
4. no real solution

Correct answer: x = -2 and x = 1

Solution

Critical points: 2x - 1 = 0 => x = 1/2; 2x + 3 = 0 => x = -3/2. Case x >= 1/2: (2x - 1) + (2x + 3) = 4x + 2 = 6 => x = 1 (valid, >= 1/2). Case -3/2 <= x < 1/2: (1 - 2x) + (2x + 3) = 4 != 6, no solution (the sum is constant 4 here). Case x < -3/2: (1 - 2x) + (-2x - 3) = -4x - 2 = 6 => -4x = 8 => x = -2 (valid, < -3/2). So x = 1 and x = -2.

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