Sure! Let's simplify the given expression:
[tex]$
\frac{3}{x-3} - \frac{5}{x+2}.
$[/tex]
### Step 1: Find a common denominator
The first step in simplifying this expression is to find a common denominator for the two fractions. The denominators are [tex]\(x - 3\)[/tex] and [tex]\(x + 2\)[/tex]. The common denominator will be the product of these two denominators:
[tex]$
(x - 3)(x + 2).
$[/tex]
### Step 2: Rewrite each fraction with the common denominator
Next, we rewrite each fraction so that they have the common denominator:
[tex]$
\frac{3}{x-3} = \frac{3(x+2)}{(x-3)(x+2)},
$[/tex]
and
[tex]$
\frac{5}{x+2} = \frac{5(x-3)}{(x+2)(x-3)}.
$[/tex]
### Step 3: Expand the numerators
Now we expand the numerators of these fractions:
[tex]$
\frac{3(x+2)}{(x-3)(x+2)} = \frac{3x + 6}{(x-3)(x+2)},
$[/tex]
and
[tex]$
\frac{5(x-3)}{(x+2)(x-3)} = \frac{5x - 15}{(x-3)(x+2)}.
$[/tex]
### Step 4: Combine the fractions
Subtract the second fraction from the first fraction, combining them over the common denominator:
[tex]$
\frac{3x + 6}{(x-3)(x+2)} - \frac{5x - 15}{(x-3)(x+2)} = \frac{(3x + 6) - (5x - 15)}{(x-3)(x+2)}.
$[/tex]
### Step 5: Simplify the numerator
Distribute the negative sign and combine like terms in the numerator:
[tex]$
(3x + 6) - (5x - 15) = 3x + 6 - 5x + 15 = -2x + 21.
$[/tex]
So, the combined fraction becomes:
[tex]$
\frac{-2x + 21}{(x-3)(x+2)}.
$[/tex]
Therefore, the simplified form of the given expression is:
[tex]$
\frac{-2x + 21}{(x-3)(x+2)}.
$[/tex]
