To solve the equation \(|2x - \frac{1}{2}| = |-1.5|\), we first recognize that the absolute value of \(-1.5\) is \(1.5\). Therefore, the equation simplifies to:

\[|2x - \frac{1}{2}| = 1.5\]

This absolute value equation can be split into two separate linear equations:

1. \(2x - \frac{1}{2} = 1.5\)
2. \(2x - \frac{1}{2} = -1.5\)

Let's solve each equation separately.

### Solving \(2x - \frac{1}{2} = 1.5\):

1. Add \(\frac{1}{2}\) to both sides:
\[
2x = 1.5 + \frac{1}{2}
\]

2. Convert \(1.5\) to a fraction to make the addition easier:
\[
1.5 = \frac{3}{2}
\]

3. Now add the fractions:
\[
2x = \frac{3}{2} + \frac{1}{2} = \frac{4}{2} = 2
\]

4. Divide both sides by 2:
\[
x = 1
\]

### Solving \(2x - \frac{1}{2} = -1.5\):

1. Add \(\frac{1}{2}\) to both sides:
\[
2x = -1.5 + \frac{1}{2}
\]

2. Convert \(-1.5\) to a fraction to make the addition easier:
\[
-1.5 = -\frac{3}{2}
\]

3. Now add the fractions:
\[
2x = -\frac{3}{2} + \frac{1}{2} = -\frac{2}{2} = -1
\]

4. Divide both sides by 2:
\[
x = -\frac{1}{2}
\]

### Solutions:

The solutions to the equation \(|2x - \frac{1}{2}| = 1.5\) are:

\[x = 1 \quad \text{and} \quad x = -\frac{1}{2}\]