To simplify the given expression [tex]\(\frac{3}{2x-1} - \frac{2}{x-4}\)[/tex], let's follow the steps for combining these two fractions and simplifying the result.
### Step-by-Step Solution:
1. Identify the common denominator:
The denominators of the two fractions are [tex]\(2x-1\)[/tex] and [tex]\(x-4\)[/tex]. To combine the fractions, we need a common denominator, which is the product of these two denominators:
[tex]\[
(2x - 1)(x - 4)
\][/tex]
2. Rewrite each fraction with the common denominator:
We need to rewrite each fraction so that they both have the common denominator [tex]\((2x - 1)(x - 4)\)[/tex].
- The first fraction:
[tex]\[
\frac{3}{2x-1} \quad \text{can be written as} \quad \frac{3(x - 4)}{(2x - 1)(x - 4)}
\][/tex]
- The second fraction:
[tex]\[
\frac{2}{x-4} \quad \text{can be written as} \quad \frac{2(2x - 1)}{(2x - 1)(x - 4)}
\][/tex]
3. Combine the fractions:
Now that both fractions have the same denominator, we can combine them:
[tex]\[
\frac{3(x - 4)}{(2x - 1)(x - 4)} - \frac{2(2x - 1)}{(2x - 1)(x - 4)}
\][/tex]
Combine the numerators:
[tex]\[
\frac{3(x - 4) - 2(2x - 1)}{(2x - 1)(x - 4)}
\][/tex]
4. Simplify the numerator:
Expand the terms in the numerator:
[tex]\[
3(x - 4) = 3x - 12
\][/tex]
[tex]\[
2(2x - 1) = 4x - 2
\][/tex]
So the numerator becomes:
[tex]\[
3x - 12 - (4x - 2) = 3x - 12 - 4x + 2 = -x - 10
\][/tex]
5. Write the final simplified expression:
Combine the numerator and the common denominator:
[tex]\[
\frac{-x - 10}{(2x - 1)(x - 4)}
\][/tex]
Therefore, the expression [tex]\(\frac{3}{2x-1} - \frac{2}{x-4}\)[/tex] simplifies to:
[tex]\[
\frac{-x - 10}{(2x - 1)(x - 4)}
\][/tex]
