Certainly! Let's simplify the given expression step-by-step:
[tex]\[ \frac{3}{2x - 1} + \frac{2 - x}{x - 2} \][/tex]
### Step 1: Simplify the second fraction
Notice that the second fraction [tex]\(\frac{2 - x}{x - 2}\)[/tex] can be simplified:
[tex]\[ \frac{2 - x}{x - 2} = \frac{-(x - 2)}{x - 2} = -1 \][/tex]
Therefore, the expression becomes:
[tex]\[ \frac{3}{2x - 1} - 1 \][/tex]
### Step 2: Rewrite the expression with a common denominator
To combine the fractions, we need a common denominator. The common denominator will be [tex]\(2x - 1\)[/tex]:
[tex]\[ \frac{3}{2x - 1} - 1 = \frac{3}{2x - 1} - \frac{2x - 1}{2x - 1} \][/tex]
### Step 3: Combine the fractions over the common denominator
Now, we combine the numerators over the common denominator:
[tex]\[ = \frac{3 - (2x - 1)}{2x - 1} \][/tex]
### Step 4: Simplify the resulting fraction
Distribute the negative sign in the numerator:
[tex]\[ = \frac{3 - 2x + 1}{2x - 1} = \frac{4 - 2x}{2x - 1} \][/tex]
### Step 5: Optionally, factor out constants
We can factor out -2 from the numerator to simplify further if desired:
[tex]\[ = \frac{-2(2x - 2)}{2x - 1} \][/tex]
However, this expression does not simplify further as a simpler form. The final simplified expression is:
[tex]\[ = \frac{4 - 2x}{2x - 1} \][/tex]
Thus, the simplified form of the given expression [tex]\(\frac{3}{2x - 1} + \frac{2 - x}{x - 2} \)[/tex] is:
[tex]\[ \boxed{\frac{4 - 2x}{2x - 1}} \][/tex]
