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.

Learn how to solve the absolute value equation |2x - 1| = 1 - 1/2x with a detailed step-by-step solution. This problem covers key concepts in absolute value equations and linear equations. Suitable for students in Grades 9-12.

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