|x + 1/2| + |x + 2| = 3x

Sure, let's solve the equation step by step.

The equation is |x + 1/2| + |x + 2| = 3x.

First, we need to consider the different cases for the absolute value.

Case 1: x + 1/2 >= 0 and x + 2 >= 0 This means x >= -1/2 and x >= -2. The intersection of these two conditions is x >= -1/2. So for x >= -1/2, the equation becomes: x + 1/2 + x + 2 = 3x 2x + 5/2 = 3x 5/2 = x

Case 2: x + 1/2 < 0 and x + 2 >= 0 This means x < -1/2 and x >= -2. The intersection of these two conditions is -2 <= x < -1/2. So for -2 <= x < -1/2, the equation becomes: -(x + 1/2) + x + 2 = 3x 1/2 + 2 = 3x 5/2 = 3x x = 5/6

Case 3: x + 1/2 < 0 and x + 2 < 0 This means x < -1/2 and x < -2. The intersection of these two conditions is x < -2. So for x < -2, the equation becomes: -(x + 1/2) - (x + 2) = 3x -2x - 5/2 = 3x -5/2 = 5x x = -1/2

So the solutions to the equation are x = 5/2, x = 5/6, and x = -1/2. However, we need to check these solutions in the original equation to make sure they are valid.

For x = 5/2, the original equation becomes: |5/2 + 1/2| + |5/2 + 2| = 3*5/2 3 + 7/2 = 15/2 15/2 = 15/2 So x = 5/2 is a valid solution.

For x = 5/6, the original equation becomes: |5/6 + 1/2| + |5/6 + 2| = 3*5/6 4/3 + 17/6 = 5/2 17/6 = 5/2 So x = 5/6 is not a valid solution.

For x = -1/2, the original equation becomes: |-1/2 + 1/2| + |-1/2 + 2| = 3*-1/2 0 + 3/2 = -3/2 -3/2 = -3/2 So x = -1/2 is a valid solution.

Therefore, the solutions to the equation are x = 5/2 and x = -1/2.