Problem 50 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{5}{2 x+8}+\frac{7}{3 x+12}$$

Step-by-Step Solution

Answer

The simplified result of the addition of the two fractions \(\frac{5}{2x+8}\) and \(\frac{7}{3x+12}\) is \(\frac{29x + 116}{(2x + 8) * (3x + 12)}\).

Step 1: Identify the denominators

The first fraction has a denominator of \(2x + 8\) and the second fraction has \(3x + 12\). It's clear from the outset, we cannot directly perform addition given the different denominators.

Step 2: Find the common denominator

We need to find a common denominator, which is a denominator that both fractions can use. The least common denominator (LCD) of \(2x + 8\) and \(3x + 12\) is the product of them, i.e., \((2x + 8) * (3x + 12)\). The fractions can then be rewritten using this new denominator.

Step 3: Adjust the numerators

Each fraction's numerator must be adjusted to account for the change in denominator. For the first fraction, the numerator is multiplied by \(3x + 12)\), which results to \(5 * (3x + 12)\). For the second fraction, the numerator is multiplied by \(2x + 8)\), resulting to \(7 * (2x + 8)\).

Step 4: Simplify each fraction

Simplify the adjusted numerators. This results in: \[\frac{15x + 60}{(2x + 8) * (3x + 12)} + \frac{14x + 56}{(2x + 8) * (3x + 12)}\]. Now, because the denominators are now the same, we can combine the numerators.

Step 5: Combine and simplify

Combine the numerators: \[\frac{(15x + 60) + (14x + 56)}{(2x + 8) * (3x + 12)}\]. Simplify the numerators to get the final result: \[\frac{29x + 116}{(2x + 8) * (3x + 12)}\].