Problem 65 - College Algebra

Question

Simplify each complex rational expression. $$\frac{x-\frac{x}{x+3}}{x+2}$$

Step-by-Step Solution

Answer

The simplified form of the complex rational expression \(\frac{x-\frac{x}{x+3}}{x+2}\) is \(\frac{x^2+2x}{(x+3)(x+2)}\).

Step 1: Rewrite The Numerator

To simplify this fraction, first, rewrite the numerator \(x-\frac{x}{x+3}\) as a single rational expression. This can be done by finding a common denominator, which in this case is \(x+3\). Rewriting \(x\) as \(\frac{x(x+3)}{x+3}\), the numerator now becomes the rational expression \(\frac{x(x+3)-x}{x+3}\) which simplifies further to \(\frac{x^2+3x-x}{x+3}\) and then to \(\frac{x^2+2x}{x+3}\).

Step 2: Substitute The Simplified Numerator In The Complex Fraction

Now, substitute the simplified expression for the numerator back into the complex fraction. The complex fraction now becomes \(\frac{\frac{x^2+2x}{x+3}}{x+2}\).

Step 3: Simplify The Complex Fraction

The entire expression is a fraction divided by a fraction, and you can simplify this by multiplying by the reciprocal. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). Therefore the complex fraction \(\frac{\frac{x^2+2x}{x+3}}{x+2}\) becomes \(\frac{x^2+2x}{x+3} \times \frac{1}{x+2}\). Now multiply across to simplify the complex rational expression to \(\frac{x^2+2x}{(x+3)(x+2)}\).