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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
\[\frac{29x+116}{(2x+8)(3x+12)}\]

Step by step solution

01

Find the LCD

The LCD of two algebraic expressions is obtained by multiplying the two denominators. In this case, the LCD of \(2x + 8\) and \(3x + 12\) is \((2x+8)(3x+12)\).
02

Rewrite the Fractions

Now, each fraction is rewritten with the LCD as new denominator. This is done by multiplying the numerator and denominator of each fraction by the missing factor in its original denominator: \[\frac{5}{2x+8} \times \frac{3x+12}{3x+12} = \frac{5(3x+12)}{(2x+8)(3x+12)}\] \[\frac{7}{3x+12} \times \frac{2x+8}{2x+8} = \frac{7(2x+8)}{(3x+12)(2x+8)}\]
03

Perform the Addition

Now, add the two fractions together: \[\frac{5(3x+12)+7(2x+8)}{(2x+8)(3x+12)}\]
04

Simplify

Simplify the above expression by carrying out the multiplication and addition in the numerator. This yields the final result: \[\frac{15x+60+14x+56}{(2x+8)(3x+12)} = \frac{29x+116}{(2x+8)(3x+12)}\]

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