Solve |x|+|x–1| = 2

Sure, let's solve the equation |x| + |x - 1| = 2 step by step.

Step 1: Consider the two cases separately, x >= 0 and x < 0.

Case 1: x >= 0 In this case, |x| = x and |x - 1| = x - 1 if x >= 1, and |x - 1| = 1 - x if x < 1.

Subcase 1.1: x >= 1 The equation becomes x + (x - 1) = 2, which simplifies to 2x - 1 = 2. Solving for x gives x = 1.5. However, this is not a valid solution because it's not in the range x >= 1.

Subcase 1.2: 0 <= x < 1 The equation becomes x + (1 - x) = 2, which simplifies to 1 = 2. This is a contradiction, so there are no solutions in this range.

Case 2: x < 0 In this case, |x| = -x and |x - 1| = 1 - x.

The equation becomes -x + (1 - x) = 2, which simplifies to -2x + 1 = 2. Solving for x gives x = -0.5. However, this is not a valid solution because it's not in the range x < 0.

So, there are no solutions to the equation |x| + |x - 1| = 2.