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10) Simplify: [tex]$\frac{5}{x+2}+\frac{6}{x+4}$[/tex]

A) [tex]$\frac{30-5x}{(x+4)(x+2)}$[/tex]

B) [tex]$\frac{9x+32-x^2}{(x+4)(x+2)}$[/tex]

C) [tex]$\frac{11x+32}{(x+4)(x+2)}$[/tex]

D) [tex]$\frac{47x+4+15x^2}{6x(x+3)}$[/tex]



Answer :

GinnyAnswer
To solve the problem [tex]\(\frac{5}{x+2} + \frac{6}{x+4}\)[/tex], we need to combine these two fractions. In order to add fractions, they must have a common denominator. Here, the common denominator is [tex]\((x+2)(x+4)\)[/tex].

First, we rewrite each fraction with the common denominator:

[tex]\[ \frac{5}{x+2} = \frac{5(x+4)}{(x+2)(x+4)} = \frac{5x + 20}{(x+2)(x+4)} \][/tex]

[tex]\[ \frac{6}{x+4} = \frac{6(x+2)}{(x+2)(x+4)} = \frac{6x + 12}{(x+2)(x+4)} \][/tex]

Next, we add these two fractions:

[tex]\[ \frac{5x + 20}{(x+2)(x+4)} + \frac{6x + 12}{(x+2)(x+4)} = \frac{(5x + 20) + (6x + 12)}{(x+2)(x+4)} \][/tex]

Combine the numerators:

[tex]\[ = \frac{5x + 6x + 20 + 12}{(x+2)(x+4)} = \frac{11x + 32}{(x+2)(x+4)} \][/tex]

Therefore, the simplified form of [tex]\(\frac{5}{x+2} + \frac{6}{x+4}\)[/tex] is:

[tex]\[ \frac{11x + 32}{(x+2)(x+4)} \][/tex]

The correct answer is:

C) [tex]\(\frac{11 x + 32}{(x+4)(x+2)}\)[/tex]

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