Problem 75 - College Algebra Essentials

Question

Perform the indicated operations. Simplify the result, if possible. $$\left(2-\frac{6}{x+1}\right)\left(1+\frac{3}{x-2}\right)$$

Step-by-Step Solution

Answer

\(\frac{2x^2 - 12}{(x+1)(x-2)}\)

Step 1: Distribute the terms

First, distribute each term of the first binomial to the terms of the second binomial: \((2 * 1) + (2 * \frac{3}{x-2}) - (\frac{6}{x+1}*1) - (\frac{6}{x+1} * \frac{3}{x-2})\)

Step 2: Simplify the expression

Now, simplify each term: \(2 + \frac{6}{x-2} - \frac{6}{x+1} - \frac{18}{(x+1)(x-2)} \)

Step 3: Create common denominators for fractions

To further simplify, each fraction will be rewritten with a common denominator, which is \((x+1)(x-2)\). The expression becomes: \(\frac{2(x+1)(x-2)}{(x+1)(x-2)} + \frac{6(x+1)}{(x+1)(x-2)} - \frac{6(x-2)}{(x+1)(x-2)} - \frac{18}{(x+1)(x-2)} \)

Step 4: Simplify terms and combine

Now, simplify the terms after multiplying and add all fractions together: \(\frac{2(x^2-x-2) + 6x -6 - 6x +12 - 18}{(x+1)(x-2)}\) . This results in \(\frac{2x^2 - 12}{(x+1)(x-2)}\)