Let's begin with recalling one of the Pythagorean Inequality Theorems.
If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is an acute triangle.
In our exercise, we are given that the longest side of a triangle has a length of 2x and the other two have lengths of 8 and 12. First of all, if 2x is the length of the longest side, then 2x is greater than the length of the longer of the other sides, 12.
2x> 12
x>6
Next, using the theorem we recalled at the beginning, we can write an inequality to find the value of x that makes this triangle acute.
( 2x)^2< 8^2+ 12^2
We will solve the inequality. Notice that x must be a positive number as 2x represents a side length. Therefore, we will consider only the positive case when taking a square root of x^2.
We got that for x<2sqrt(13) the triangle is acute. However, we must remember the first condition x>6. Therefore, the answer will be a compound inequality.
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