1. In the triangle shown, the side lengths are given below in terms of a real-valued variable $x$. Find the range of all possible values of $x$, writing your answer in interval notation.
https://latex.artofproblemsolving.com/6/c/a/6ca2d738712c186c784acd9ae288810db5edaadd.png
2.One side of a triangle has length $10$, and the triangle has integer perimeter. What is the smallest possible perimeter of the triangle?
3.Two side lengths of a triangle are $11$ and $17$. What is the longest possible integer length of the third side of the triangle?
1. The set of possible x is [3,25].
2. The smallest possible perimeter is 23.
3. The largest possible third side is 29.
+23252 I'm not certain what the "$" on each side of "x" represent.
If you are just considering all real numbers for the possibile replacements for x in the triangle whose sides are x + 15, 2x + 15, and 4x + 15, then, since the sum of any two sides of a triangle must be greater than the third side, there are these three possibilities:
I. (x + 15) + (2x + 15) > 4x + 15 ---> 3x + 30 > 4x + 15 ---> 15 > x ---> x < 15
II. (x + 15) + (4x + 15) > 2x + 15 ---> 5x + 30 > 2x + 15 ---> 3x > -15 ---> x > -5
III. (2x + 15) + (4x + 15) > x + 15 ---> 6x + 30 > x + 15 ---> 5x > -15 ---> x > -3
The answer is (-3, 15) [Using a value of -4 or -5 for x results in a side of the triangle being negative.]
Using "$x$ to refer to just "x", I can't get any possible of answers that work for questions 2 and 3.
