Problem 75 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+3}{3 x+6}+\frac{x}{4-x^{2}}$$

Step-by-Step Solution

Answer

The simplified result of the sum is \(\frac{6 - x^2}{3(2 + x)(2 - x)}\)

Step 1: Find a common denominator

First, factorize each denominator. The first denominator \(3x+6\) can be simplified to \(3(x+2)\), and the second denominator \(4-x^2\) can be factorized using the difference of squares formula as \((2+x)(2-x)\). The least common denominator (LCD) of these two fractions would be \(3(2+x)(2-x)\)

Step 2: Rewrite the fractions with the common denominator

Next, rewrite each fraction with the common denominator: \[\frac{x + 3}{3(x + 2)} = \frac{(x + 3)(2 - x)}{3(2 + x)(2 - x)}\]and \[\frac{x}{4 - x^2} = \frac{3x}{3(2 + x)(2 - x)}\]

Step 3: Combine and simplify

Now, add the two fractions together: \[\frac{(x + 3)(2 - x)}{3(2 + x)(2 - x)} + \frac{3x}{3(2 + x)(2 - x)}\]The numerators combine to \((x + 3)(2 - x) + 3x\). Simplify this to \(6 - x^2\). Therefore, the simplified fraction becomes \[\frac{6 - x^2}{3(2 + x)(2 - x)}\]