Problem 76 - Introductory Algebra for College Students

Question

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x+7}{4 x+12}+\frac{x}{9-x^{2}}$$

Step-by-Step Solution

Answer

The simplified sum of the two fractions is \[\frac{(x+7)(9-x^{2}) + x(4x+12)}{(4x+12)(9-x^{2})}\]

Step 1: Identify Common Denominator

In this scenario, the common denominator is the least common multiple (LCM) of \(4x + 12\) and \(9 - x^{2}\). The LCM for these two expressions is their product, i.e., \((4x + 12)(9 - x^{2})\) since they don't have common root.

Step 2: Rewrite Fractions

Rewrite each fraction with the new denominator: \[\frac{x+7}{4x+12} * \frac{9-x^{2}}{9-x^{2}} = \frac{(x+7)(9-x^{2})}{(4x+12)(9-x^{2})}\] and \[\frac{x}{9-x^{2}} * \frac{4x+12}{4x+12} = \frac{x(4x+12)}{(4x+12)(9-x^{2})}\]

Step 3: Perform Addition

Since these fractions now have the same denominator, they can be added directly: \[\frac{(x+7)(9-x^{2}) + x(4x+12)}{(4x+12)(9-x^{2})}\]

Step 4: Simplify

The expression can then be simplified (if applicable). In this case, it is not possible to simplify any further, since no factors are common to the numerator and the denominator.