Question

Subtract the rational expressions and simplify your final answer.
6x-3/4x^2 -9 - 2/2x+3

Answer 1

To subtract the rational expressions \(\frac{6x - 3}{4x^2 - 9} - \frac{2}{2x + 3}\), we first need to get a common denominator.

### Step 1: Factor the denominators

1. The denominator \(4x^2 - 9\) can be factored as a difference of squares:
\[
4x^2 - 9 = (2x - 3)(2x + 3)
\]

2. The second denominator \(2x + 3\) is already in factored form.

### Step 2: Determine the common denominator

The common denominator for the two expressions will be:
\[
(2x - 3)(2x + 3)
\]

### Step 3: Rewrite each expression with the common denominator

1. For the first expression:
\[
\frac{6x - 3}{(2x - 3)(2x + 3)}
\]
It is already in the correct form.

2. For the second expression \(\frac{2}{2x + 3}\), we need to multiply the numerator and denominator by \(2x - 3\):
\[
\frac{2}{2x + 3} \cdot \frac{2x - 3}{2x - 3} = \frac{2(2x - 3)}{(2x - 3)(2x + 3)} = \frac{4x - 6}{(2x - 3)(2x + 3)}
\]

### Step 4: Combine the fractions

Now we rewrite the subtraction:
\[
\frac{6x - 3}{(2x - 3)(2x + 3)} - \frac{4x - 6}{(2x - 3)(2x + 3)} = \frac{(6x - 3) - (4x - 6)}{(2x - 3)(2x + 3)}
\]

### Step 5: Simplify the numerator

Now simplify the numerator:
\[
(6x - 3) - (4x - 6) = 6x - 3 - 4x + 6 = 2x + 3
\]

### Step 6: Combine everything

Now we have:
\[
\frac{2x + 3}{(2x - 3)(2x + 3)}
\]

### Step 7: Simplify the expression

Notice that \(2x + 3\) in the numerator and denominator can be canceled:
\[
\frac{2x + 3}{(2x - 3)(2x + 3)} = \frac{1}{2x - 3} \quad (\text{provided } 2x + 3 \neq 0)
\]

### Final Answer

Thus, the simplified expression is:
\[
\frac{1}{2x - 3}
\]