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Chapter 0: Problem 65

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

Short Answer

Expert verified
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 by step solution

01

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 \(\frac{x^2+2x}{x+3}\).
02

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 \(\frac{\frac{x^2+2x}{x+3}}{x+2}\).
03

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 \(\frac{x^2+2x}{(x+3)(x+2)}\).

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