What does the equation |x + 1| = |2x - 1| yield for x?

To solve the equation ( |x + 1| = |2x - 1| ), we need to consider the properties of absolute values. The equation can be interpreted in terms of two cases based on the definition of absolute value.

1. Case 1: The expressions inside the absolute values are both non-negative or both non-positive. This leads us to set up the equation without the absolute value signs:

[

x + 1 = 2x - 1

]

Solving this, we rearrange the terms:

[

1 + 1 = 2x - x

]

[

2 = x

]

2. Case 2: One expression is the negative of the other. This leads to:

[

x + 1 = -(2x - 1)

]

Simplifying gives:

[

x + 1 = -2x + 1

]

Rearranging yields:

[

x + 2x = 1 - 1

]

[

3x = 0 \implies x =