Question

find the perimeter of the rectangle.
\\( \frac { 3 } { x - 2 } \\)
\\( \frac { 4 } { x + 1 } \\)

Explanation:

Step1: Recall the formula for the perimeter of a rectangle

The perimeter \(P\) of a rectangle is \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. Here, \(l=\frac{3}{x - 2}\) and \(w=\frac{4}{x + 1}\).

Step2: Substitute into the perimeter formula

\(P=2(\frac{3}{x - 2}+\frac{4}{x + 1})\)

Step3: Find a common denominator

The common denominator of \(x - 2\) and \(x + 1\) is \((x - 2)(x + 1)\). So \(\frac{3}{x - 2}+\frac{4}{x + 1}=\frac{3(x + 1)+4(x - 2)}{(x - 2)(x + 1)}\)
Expand the numerator: \(3(x + 1)+4(x - 2)=3x+3 + 4x-8=7x - 5\)

Step4: Simplify the perimeter expression

\(P = 2\times\frac{7x - 5}{(x - 2)(x + 1)}=\frac{14x - 10}{(x - 2)(x + 1)}=\frac{14x - 10}{x^{2}-x - 2}\)

Answer:

\(\frac{14x - 10}{x^{2}-x - 2}\)