The sides of an acute triangle are x, 35, and 37, where x is the shortest side. Find the range of possible values for x and express it as an inequality
To find the range of possible values for x, the shortest side, in an acute triangle with side lengths x, 35, and 37, we need to apply the Triangle Inequality Theorem. According to this theorem, in any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
In this case, we can apply the theorem to the given sides:
x + 35 > 37 (the sum of x and 35 must be greater than 37)
x + 37 > 35 (the sum of x and 37 must be greater than 35)
35 + 37 > x (the sum of 35 and 37 must be greater than x)
Simplifying these inequalities, we have:
x > 2 (subtracting 35 from both sides)
x > -2 (subtracting 37 from both sides)
72 > x (subtracting 35 from both sides)
From these inequalities, we can see that x must be greater than 2, greater than -2, and less than 72. Therefore, the range of possible values for x can be expressed as:
-2 < x < 72
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