Question

express the perimeter of the rectangle as a single rational expression
what is the perimeter, p, of the rectangle?
p = (simplify your answer.)

Explanation:

Step1: Recall perimeter formula

The perimeter formula of a rectangle is $P = 2(l + w)$, where $l$ is the length and $w$ is the width. Here, $l=\frac{x}{x + 6}$ and $w=\frac{x}{x + 5}$. So $P=2(\frac{x}{x + 6}+\frac{x}{x + 5})$.

Step2: Find common - denominator

The common denominator of $\frac{x}{x + 6}$ and $\frac{x}{x + 5}$ is $(x + 6)(x + 5)$. Then $\frac{x}{x + 6}+\frac{x}{x + 5}=\frac{x(x + 5)+x(x + 6)}{(x + 6)(x + 5)}=\frac{x^{2}+5x+x^{2}+6x}{(x + 6)(x + 5)}=\frac{2x^{2}+11x}{(x + 6)(x + 5)}$.

Step3: Multiply by 2

$P = 2\times\frac{2x^{2}+11x}{(x + 6)(x + 5)}=\frac{4x^{2}+22x}{(x + 6)(x + 5)}$.

Answer:

$\frac{4x^{2}+22x}{(x + 6)(x + 5)}$