Solve the Absolute Value Equation: Balancing |x-1| and |2x+3|

To solve the equation $|x-1| = |2x+3|$, follow these steps:

• First, recall the property: for any real numbers $a$ and $b$, $|a| = |b|$ implies $a = b$ or $a = -b$.
• We will consider two cases based on this property:

Case 1: Assume $x-1 = 2x+3$.
Simplify the equation:
$x-1 = 2x+3$
Subtract $x$ from both sides:
$-1 = x+3$
Subtract 3 from both sides:
$x = -4$.

Case 2: Assume $x-1 = -(2x+3)$.
Simplify the equation:
$x-1 = -2x-3$
Add $2x$ to both sides:
$3x-1 = -3$
Add 1 to both sides:
$3x = -2$
Divide everything by 3:
$x = -\frac{2}{3}$.

Therefore, the solutions to the equation $|x-1| = |2x+3|$ are $x = -4$ and $x = -\frac{2}{3}$.

These solutions correspond to answer choice 4: $x = -4$, $x = -\frac{2}{3}$.

Thus, $x = -4$ and $x = -\frac{2}{3}$.