Question P.6.8: Subtracting Rational Expressions with Different Denominators...

Step 1   Find the least common denominator. In Example 6, we found that the LCD for these rational expressions is (2x – 3)(x + 3).

Step 2   Write equivalent expressions with the LCD as denominators. We must rewrite each rational expression with a denominator of (2x – 3)(x + 3). We do so by multiplying both the numerator and the denominator of each rational expression by any factor needed to convert the expression’s denominator into the LCD.

Because \frac{x+3}{x+3}=1 and \frac{2x-3}{2x-3}=1, we are not changing the value of either rational expression, only its appearance.

Now we are ready to perform the indicated subtraction.

\frac{x + 2}{2x – 3} – \frac{4}{x + 3}
This is the given problem.

=\frac{(x + 2)(x+3)}{(2x – 3)(x+3)} – \frac{4(2x-3)}{(x + 3)(2x-3)}
Multiply each numerator and denominator by the extra factor required to form (2x – 3)(x + 3), the LCD.

Step 3   Subtract numerators, putting this difference over the LCD.

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

=\frac{x² + 5x + 6 – (8x – 12)}{(2x – 3)(x + 3)}
Multiply in the numerator using FOIL and the distributive property.

=\frac{x² + 5x + 6 – 8x + 12}{(2x – 3)(x + 3)}
Remove parentheses and then change the sign of each term in parentheses.

=\frac{x² – 3x + 18}{(2x – 3)(x + 3)},  x ≠ \frac{3}{2},  x ≠ -3
Combine like terms in the numerator.

Step 4 If necessary, simplify. Because the numerator is prime, no further simplification is possible.