To solve the equation $|2x - 1| = 1 - \frac{1}{2}x$, we'll need to consider two cases based on the definition of the absolute value function.

Case 1: $2x - 1 \geq 0$

In this case, the absolute value expression $|2x - 1|$ simplifies to $2x - 1$. So the equation becomes:

$2x - 1 = 1 - \frac{1}{2}x$

Now, solve for $x$:

1. Add $\frac{1}{2}x$ to both sides: $2x + \frac{1}{2}x - 1 = 1$ Combine like terms: $\frac{5}{2}x - 1 = 1$
2. Add 1 to both sides: $\frac{5}{2}x = 2$
3. Multiply both sides by $\frac{2}{5}$: $x = \frac{4}{5}$

Check whether this solution satisfies the condition $2x - 1 \geq 0$: $2\left(\frac{4}{5}\right) - 1 = \frac{8}{5} - \frac{5}{5} = \frac{3}{5} \geq 0$ Since the condition is satisfied, $x = \frac{4}{5}$ is a valid solution for this case.

Case 2: $2x - 1 < 0$

In this case, the absolute value expression $|2x - 1|$ simplifies to $-(2x - 1)$. So the equation becomes:

$-(2x - 1) = 1 - \frac{1}{2}x$

Now, solve for $x$:

1. Distribute the negative sign: $-2x + 1 = 1 - \frac{1}{2}x$
2. Add $2x$ to both sides: $1 = 2x + 1 - \frac{1}{2}x$ Combine like terms: $1 = \frac{3}{2}x + 1$
3. Subtract 1 from both sides: $0 = \frac{3}{2}x$
4. Multiply both sides by $\frac{2}{3}$: $x = 0$

Check whether this solution satisfies the condition $2x - 1 < 0$: $2(0) - 1 = -1 < 0$ Since the condition is satisfied, $x = 0$ is a valid solution for this case.

Final Solution:

The solutions are $x = \frac{4}{5}$ and $x = 0$.

Would you like any further details or have any questions? Here are some related problems:

1. Solve $|3x + 4| = 2x - 5$.
2. Solve $|5x - 2| = 3x + 1$.
3. Solve $|x - 1| = 2x + 3$.
4. Solve $|4x + 3| = 2x + 5$.
5. Solve $|7x - 4| = 3 - 2x$.

Tip: When solving absolute value equations, always consider both the positive and negative cases of the expression within the absolute value.