We need to evaluate the expression $-x - \frac{1}{2}x + \frac{3}{2}$ when $x = -\frac{1}{4}$.

1. Problem Description

We need to evaluate the expression $-x - \frac{1}{2}x + \frac{3}{2}$ when $x = -\frac{1}{4}$.

2. Solution Steps

First, substitute $x = -\frac{1}{4}$ into the expression:

$-x - \frac{1}{2}x + \frac{3}{2} = -(-\frac{1}{4}) - \frac{1}{2}(-\frac{1}{4}) + \frac{3}{2}$

Simplify the terms:

$-(-\frac{1}{4}) = \frac{1}{4}$

$-\frac{1}{2}(-\frac{1}{4}) = \frac{1}{8}$

Now substitute these values back into the expression:

$\frac{1}{4} + \frac{1}{8} + \frac{3}{2}$

Find a common denominator, which is 8:

$\frac{1}{4} = \frac{2}{8}$

$\frac{3}{2} = \frac{12}{8}$

Now add the fractions:

$\frac{2}{8} + \frac{1}{8} + \frac{12}{8} = \frac{2+1+12}{8} = \frac{15}{8}$

3. Final Answer

$\frac{15}{8}$