The sides of a pentagon are represented in centimeters by $x$, $10$, $2x$, $1$ and $3$. How many even values of $x$ ​satisfy this pentagon?

Question

The sides of a pentagon are represented in centimeters by $x$, $10$, $2x$, $1$ and $3$. Determine how many even values of $x$ ​​are there that satisfy this pentagon. The answer is 5.

How can I solve this problem? Is the triangle inequality useful, because when I used it, I couldn't find the range for $x$.

Can someone help me?

Answer 1

In order for the sides to form a pentagon, the longest side has to be less than the sum of the other sides (this is equivalent to triangle inequality). You have two cases:

1. The longest side is $10$, so you can write $$10<x+2x+1+3$$ or $x>2$

2. The longest side is $2x$, so $$2x<x+10+1+3$$ or $x<14$

Therefore, the allowed values are $4,6,8,10,12$