Question

The lengths of two sides of a triangle are 10 and 24, and the third side is x. How many whole number values are possible for x.

Answer 1

`20`.

Explanation:

In any triangle, the sum of the lengths of any two sides should be strictly greater than the length of the third side. For example, if the length of the three sides are
`a`,
`b`, and
`c`:

`a + b > c`,

`a + c > b`, and

`b + c > a`.

In this question, the length of the sides are
`10`,
`24`, and
`x`. The length of these sides should satisfty the following inequalities:

`10 + 24 > x`,

`10 + x > 24`, and

`24 + x > 10`.

Since
`x > 0`, the inequality
`24 + x > 10` is guarenteed to be satisfied.

Simplify
`10 + 24 > x` to obtain the inequality
`x < 35`.

Similarly, simplify
`10 + x > 24` to obtain the inequality
`x > 14`.

Since
`x` needs to be a whole number, the greatest
`x` that satisfies
`x < 35` would be
`34`. Similarly, the least
`x\!` that satisfies
`x > 14` would be
`15`. Thus,
`x\!\!` could be any whole number between
`15\!` and
`34\!` (inclusive.)

There are a total of
`34 - 15 + 1 = 20` distinct whole numbers between
`15` and
`34` (inclusive.) Thus, the number of possible whole number values for
`x` would be
`20`.