Question

The sum of the measures of any two sides of a triangle must be greater than the measure of the third side. What is the greatest possible integer value for x?

Answer 1

try using Pythagorean theorem

Answer 2

"The sum of the measures of any two sides of a triangle must be greater than the measure of the third side"
You should state the remainder of that theorem:
"The length of a side of a triangle is less than the sum of the lengths of the other two sides
AND greater than the difference of the lengths of the other two sides."
By stating the entire rule, you can exclude the existence of a triangle of sides 9, 3 and 2.
Third side must be < sum of any two sides and
Third Side must be > difference of lengths of any 2 sides
So, if we have 2 sides of a triangle that are 15 and 12, the third side has to be less than (15 + 12) and greater than (15 -12). So side x must be >3 and <27.
To answer your question, side x cannot be 27 so the largest integer value it can have is 26.