A triangle has side lengths 2x+2ft, x+3 ft and n ft.

Which expression represents the possible values of n, in feet?

A) x-1 < n < 3x+5

B)n=3x+5

C)n=x-1

D)3x+5 < n < x-1

  A)  x-1 < n < 3x+5

Step-by-step explanation:

The value of n can range between the sum and difference of the lengths of the other two sides. The sum is ...

   (2x +2) +(x +3) = 3x +5

The difference is ...

  (2x +2) -(x +3) = x -1

For the purpose of choosing one of these answers, we must assume that the sum is greater than the difference and the value of x is such that x-1 > 0. Using these assumptions, possible values of n are ...

  x -1 < n < 3x +5 . . . . . for x > 1

Alternate Solution

The expressions for the given side lengths are both positive when x > -1. In the range -1 < x < 1, we have the condition that 2x+2 ranges from 0 to 4 and x+3 ranges from 2 to 4. That is (x+3) > (2x+2) and possible values of n are ...

   lowest: (x+3) -(2x +2) = 1 -x

   highest: (x+3) +(2x +2) = 3x +5

So, another possible solution is ...

  1-x < n < 3x +5 . . . . . .  . for -1 < x < 1