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Chapter 0: Problem 50

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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 \(7 * (2x + 8)\).
04

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.
05

Combine and simplify

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

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