Problem 75 - College Algebra

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

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

Step 1 - 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: \(2*(1) + 2*(\frac{3}{x-2}) - \frac{6}{x+1}*(1) - \frac{6}{x+1}*(\frac{3}{x-2})\).

Step 2 - Simplify the expression

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

Step 3 - Find Common denominator

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

Step 4 - Simplify the Numerator

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

Step 5 - 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 \(2\).