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Solve for x: |2x - 3| + |2x + 1| + |2x + 5| = 12.

1. x = -5/2 or x = 3/2
2. x = -3/2 or x = 3/2
3. x = -1/2 or x = 5/2
4. x = 0 or x = 1

Correct answer: x = -5/2 or x = 3/2

Solution

Critical points: 2x-3=0 => x=3/2; 2x+1=0 => x=-1/2; 2x+5=0 => x=-5/2. The minimum value of the sum (at the median point x=-1/2) is |−4|+0+|4| = 8 < 12, so two solutions lie in the outer regions. For x >= 3/2: (2x-3)+(2x+1)+(2x+5) = 6x+3 = 12 => x = 3/2. For x <= -5/2: -(2x-3)-(2x+1)-(2x+5) = -6x-3 = 12 => x = -5/2.

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