Actually, that's it. Why did the case of 1 get the correct result when using cosine theorem to find obtuse triangle? In the second case, when finding an acute triangle, the result is incorrect.

First, the long side faces a large angle. Therefore, in an obtuse triangle, the longest side corresponds to the largest angle-obtuse angle. In your first example, the trilateral relationship has been determined, which is 2a+ 1. So the angle corresponding to this side must be obtuse. Then the range of a can be found according to the cosine of this angle. It should be a>2 and 1

But in the second case, the size relationship between x and 3 is uncertain. So it is uncertain whether the angle corresponding to the X side is larger or the angle corresponding to the 3 is larger. An acute triangle must have three acute angles; That is, the maximum angle must be acute. Therefore, it is necessary to use the first cosine theorem for the acute angle of the side corresponding to X and the first cosine theorem for the side corresponding to 3. The combination of the results obtained from two inequalities is the desired result.

However, if the range of X is found similar to the obtuse triangle with sides of 2, 3 and X in the second example, if the angle corresponding to X is found simply by cosine theorem, the result will be incorrect. Because it is possible that 3 corresponds to an obtuse angle.

So your summary is incorrect. 1 The reason why the result is correct is not to find an obtuse triangle, but to determine the size relationship of three sides. The reason for the second force error is not to find an acute triangle, but that the relationship between the two sides 3 and X is uncertain.