Problem 70 - College Algebra

Question

Simplify each complex rational expression. $$\frac{\frac{6}{x^{2}+2 x-15}-\frac{1}{x-3}}{\frac{1}{x+5}+1}$$

Step-by-Step Solution

Answer

The simplified expression is \(-\frac{x-7}{x-3}\).

Step 1: Identify the individual fractions

First, recognize the individual fractions in the numerator and the denominator of the complex expression: \(\frac{6}{x^{2}+2x-15}\), \(\frac{1}{x-3}\), \(\frac{1}{x+5}\), and \(1\).

Step 2: Combine fractions in the numerator

To combine the fractions in the numerator, we need a common denominator: \(x^{2}+2x-15\) and \(x-3\), which factors into \((x+5)\(x-3)\). The combined fraction at the numerator then becomes: \(\frac{6 - (x^{2}+2x-15)}{(x-3)(x+5)} = \frac{-x^{2}+4x+21}{x^{2}-2x-15}\).

Step 3: Combine fractions in the denominator

To combine the fractions in the denominator, we need a common denominator as well. In this case, it's \(x+5\). When we combine the fractions then, we'll have: \(\frac{1+ x+5}{x+5} = \frac{x+6}{x+5}\).

Step 4: Invert and multiply

To divide the fractions, change the division to multiplication and flip (take the reciprocal of) the second fraction, which means that we multiply the numerator by the reciprocal of the denominator: \((-x^{2}+4x+21)/(x^{2}-2x-15)\) * \((x+5)/(x+6)\).

Step 5: Factor and simplify

Now we can simplify by factoring the expressions and cancelling out common factors. The equation is rewritten as \(\frac{-(x+3)(x-7)}{(x-3)(x+5)}\) * \(\frac{(x+5)}{(x+6)}\) which simplifies to \(-\frac{x-7}{x-3}\) when you cancel out the common factors.

Step 6: Final answer

The final simplified expression is: \(-\frac{x-7}{x-3}\).