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

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

• A. x = 0 and 2
• B. x = -1 and 1
• C. x = 1 and 3
• D. x = -2 and 2

Answer: (A)

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 =

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 =