two sides of a triangle have lengths 45 inches and 50 inches. How long can the third side be?

any side must be less than the sum of the other two sides.

So, it must be less than 95.

I could be any number between 5 and 95 inches

let it be x
x+ 45 >50 ---> x > 5
x+50 > 45 -- x > -5
50 + 45 > x
x < 95

so x > -5 and x > 5 and x < 95
so 5 < x < 95

To find the maximum length of the third side in a triangle, we can apply the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

So, for a triangle with sides measuring 45 inches and 50 inches, we need to find the range of values that the third side can fall within.

Let's consider the two cases:

1. The sum of the two given sides is less than the length of the missing side:
In this case, the missing side cannot connect the two given sides to form a triangle. Therefore, the length of the third side will be less than the difference between the lengths of the two given sides.

Third side < |length of first side - length of second side|

Third side < |45 - 50|
Third side < |-5|
Third side < 5 inches

2. The sum of the two given sides is equal to or greater than the length of the missing side:
In this case, the missing side can potentially connect the two given sides to form a triangle. Therefore, the length of the third side should be less than the sum of the lengths of the two given sides.

Third side < length of first side + length of second side

Third side < 45 + 50
Third side < 95 inches

Combining both cases, we can conclude that the length of the third side can be any value between 5 inches exclusive and 95 inches inclusive.