Problem 40 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{4 x}{x^{2}-25}+\frac{x}{x+5}$$

Step-by-Step Solution

Answer

The result after simplifying the expression is \(\frac{x \cdot (x-1)}{(x-5)(x+5)}\).

Step 1: Identify the denominators

Identify the denominators of the two fractions, which are \(x^{2}-25\) and \(x+5\).

Step 2: Finding the Least Common Denominator (LCD)

In order to add or subtract fractions, we need to have the same denominators. Here, we can factor the first denominator to make it look like the second denominator. \(x^{2}-25\) is a difference of squares and can be factored as \((x-5)(x+5)\).

Step 3: Rewrite the fractions

Now we can write both fractions with the same denominator. Thus, the original equation becomes: \(\frac{4 x}{(x-5)(x+5)}+\frac{(x)(x-5)}{(x-5)(x+5)}\)

Step 4: Simplify

We can combine the fractions and simplify. Continue to simplify by combining like terms in the numerator: \(\frac{4 x+x(x-5)}{(x-5)(x+5)}=\frac{4x+x^{2}-5x}{(x-5)(x+5)}=\frac{x^{2}-x}{(x-5)(x+5)}\)

Step 5: Further simplification

We can simplify the result by factoring the numerator. Factoring \(x^{2}-x\) we get \(x \cdot (x-1)\), so the final answer is \(\frac{x \cdot (x-1)}{(x-5)(x+5)}\)