Problem 42 - College Algebra

Question

Add or subtract as indicated. $$\frac{8}{x-2}+\frac{2}{x-3}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{5x-14}{(x-2)(x-3)}\).

Step 1: Find the Least Common Denominator (LCD)

Firstly, the Least Common Denominator (LCD) needs to be found. In this case, since the denominators \(x-2\) and \(x-3\) are both different and there are no common factors, the LCD will be the product of the two denominators. Therefore, the LCD is \((x-2)(x-3)\).

Step 2: Rewrite the fractions with the LCD

Rewrite the fractions using the LCD. This is done by multiplying the numerator and denominator of the first fraction by \(x-3\) and the numerator and denominator of the second fraction by \(x-2\). The expression becomes: \(\frac{8(x-3)}{(x-2)(x-3)} + \frac{2(x-2)}{(x-2)(x-3)}\).

Step 3: Simplify the fractions

Now, simply add the two fractions together (since they have the same denominator, they can be added directly). The resulting fraction is: \(\frac{8(x-3)+2(x-2)}{(x-2)(x-3)}\). Distribute the numerators to get \(\frac{8x-24+2x-4}{(x-2)(x-3)}\). This simplifies further to: \(\frac{10x-28}{(x-2)(x-3)}\) or further simplified to \(\frac{5x-14}{(x-2)(x-3)}\).