Problem 42

Question

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

Step-by-Step Solution

Verified

Answer

The sum of the fractions $\frac{8}{x-2}+\frac{2}{x-3}$ is $10x-28$.

1Step 1: Find the Least Common Denominator (LCD)

In order to add or subtract fractions, they must have the same denominator or in other words, we need a least common denominator (LCD). In this case, the LCD of $x-2$ and $x-3$ is their product which is $(x-2)(x-3)$.

2Step 2: Rewrite the fractions

Rewrite fractions using the LCD. In this case, the equation turns into $\frac{8}{x-2}*(x-3)+(x-2)*\frac{2}{x-3}$. The objective here is to make the denominators the same.

3Step 3: Simplification

Simplify the resulting expression by multiplying the numerators and getting rid of the denominators. It will result in $8*(x-3)+2*(x-2)$.

4Step 4: Distribution

Distribute 8 in $8*(x-3)$ and 2 in $2*(x-2)$, we get $8x-24+2x-4$.

5Step 5: Combine like terms

Combine terms to obtain the final result. The equation simplifies to $10x-28$ once like terms are combined.

Problem 42