Problem 50 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \((2x+8)(3x+12)\).

Step 2: 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: \[\frac{5}{2x+8} \times \frac{3x+12}{3x+12} = \frac{5(3x+12)}{(2x+8)(3x+12)}\] \[\frac{7}{3x+12} \times \frac{2x+8}{2x+8} = \frac{7(2x+8)}{(3x+12)(2x+8)}\]

Step 3: Perform the Addition

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

Step 4: Simplify

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