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Chapter 0: Problem 85

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

Short Answer

Expert verified
The final simplified result for this expression is \(\frac{2x^2 + 6x - 18}{(x+1)(x-2)}\)

Step by step solution

01

Distribute Each Term

Initially, each term in the first binomial needs to be distributed across each term in the second binomial. The operation will look like: \(2 * (1 + \frac{3}{x-2}) - \frac{6}{x+1} * (1 + \frac{3}{x-2})\)
02

Simplify by Distribution

Carry out the multiplication in each term to simplify: \(2 + \frac{6}{x-2} - \frac{6}{x+1} - \frac{18}{(x+1)(x-2)}\)
03

Common Denominator

Next, get a common denominator for all the fractions to combine them. In this case, the common denominator is \((x+1)(x-2)\). After expanding the terms and simplifying, the expression becomes: \(\frac{2(x+1)(x-2) + 6(x+1) - 6(x-2) - 18}{(x+1)(x-2)}\)
04

Final Simplification

Finally, simplify the expression in the numerator to get the simplest form which should be: \(\frac{2x^2 - 4x + 6x - 12 + 6x - 12 - 6x + 12 - 18}{(x+1)(x-2)} = \frac{2x^2 + 6x - 18}{(x+1)(x-2)}\)

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