Write an equation for the perimeter of a rectangle and solve for $x$ given dimensions and perimeter

Question

I found an interesting math problem. The perimeter of rectangle is 24 cm, and its dimensions are 2/x and 5/(x+1).
Form an equation and find the value of x. I got stuck with 2 values of x: x=-2/3 or x=1/4. I cannot understand which one to use because if we can use both of them, one side will have a negative length.

Answer 1

The perimeter is given by

$$P=2(L+l)=2(\frac{5}{x+1}+\frac 2x)=24$$ with $x>0$

thus

$$\frac{5}{x+1}+\frac 2x =12$$

and

$$5x+2(x+1)=12x(x+1)$$ then $$12x^2+12x-5x-2x-2=0$$ or $$12x^2+5x-2=0$$ $$\delta=25+96=121$$ the positive root is $$x=\frac{-5+11}{24}=\frac 14$$

Then $$L=\frac 2x=8\; \text{ and } \; l=4$$

Answer 2

Substituting the first solution $(x=-2/3)$ back into the length of the rectangle, we get $l=-2/x=-3$. This is clearly not possible as the length can never be negative. Hence the first solution is invalid and can be neglected.

Answer 3

The length of the sides are given by $\frac2x, \frac5{x+1}$, and these must be positive because lengths are non-negative.

Therefore you have been asked to solve the system of equations:

$\frac2x\gt0, \frac5{x+1}\gt0, \frac2x+\frac5{x+1}=12$, and the answer is $x=\frac14$.