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Chapter 0: Problem 75

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 simplified result of the expression \((2-\frac{6}{x+1})(1+\frac{3}{x-2})\) is \(2\).

Step by step solution

01

- Distribute the left and right terms

Distribute the terms by multiplying each term of \((2-\frac{6}{x+1})\) with each term of \((1+\frac{3}{x-2})\). This results in: \(2*(1) + 2*(\frac{3}{x-2}) - \frac{6}{x+1}*(1) - \frac{6}{x+1}*(\frac{3}{x-2})\).
02

- Simplify the expression

Simplify the expression as: \(2 + \frac{6}{x-2} - \frac{6}{x+1} - \frac{18}{(x+1)(x-2)}\).
03

- Find Common denominator

Make the denominators same, because, to add or subtract fractions, the denominators must be the same: \(2 + \frac{6(x+1) - 6(x-2) - 18}{(x+1)(x-2)}\).
04

- Simplify the Numerator

Simplify the numerator as: \(2+ \frac{(6x+6 - 6x +12 - 18)}{(x+1)(x-2)}\).
05

- Final Simplification

Simplify the final expression as: \(2+ \frac{0}{(x+1)(x-2)}\). Since zero divided by anything is still zero, this simplifies to \(2+0\) or just \(2\).

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