Okay, so I need to figure out the possible integer length of the third side of a triangle when two of the sides are 1 unit and 3 units. Hmm, let me start by recalling the triangle inequality theorem. That's the one that says the sum of any two sides of a triangle must be greater than the third side. Right. So, if the sides are a, b, and c, then a + b > c, a + c > b, and b + c > a. 

Alright, let's apply that here. The two given sides are 1 and 3, and the third side is an integer we need to find. Let's denote the third side as x. So, the sides are 1, 3, and x. Now, according to the triangle inequality:

1. 1 + 3 > x
2. 1 + x > 3
3. 3 + x > 1

Let me simplify each of these inequalities.

Starting with the first one: 1 + 3 > x, which simplifies to 4 > x or x < 4.

Second inequality: 1 + x > 3. Subtracting 1 from both sides gives x > 2.

Third inequality: 3 + x > 1. Subtracting 3 from both sides gives x > -2. But since a side length can't be negative, this doesn't really affect our possible values. So, we can ignore this third inequality because x has to be positive anyway.

So combining the first two inequalities, we have x > 2 and x < 4. Since x must be an integer, the possible values of x are 3. Because 2 < x < 4, and x is an integer, so x must be 3. Wait, but let me check if 3 satisfies all the inequalities.

Check the first inequality: 1 + 3 = 4, which is greater than 3 (since 4 > 3) ✔️.

Second inequality: 1 + 3 = 4, which is greater than 3 (4 > 3) ✔️.

Third inequality: 3 + 3 = 6, which is greater than 1 (6 > 1) ✔️.

So yes, x = 3 is a valid length. But wait, is there another possible integer? Let's see. If x were 2, then the sides would be 1, 3, 2. Let's check the triangle inequalities:

1 + 2 > 3? 3 > 3? No, that's not true. 3 is not greater than 3. So 2 is invalid. What about x = 4? Then sides would be 1, 3, 4. Check the inequalities:

1 + 3 = 4, which is not greater than 4. So that fails as well. So x can't be 4. Therefore, the only integer that works is 3.

Wait, but let me think again. The problem says the third side is an integer value. So, according to the triangle inequality, x must be greater than 2 and less than 4. The only integer in that range is 3. So the answer should be 3.

But just to make sure I didn't miss anything. Let me visualize the triangle. If the sides are 1, 3, and 3, does that form a valid triangle? Yes, because 1 + 3 > 3 (4 > 3), 3 + 3 > 1 (6 > 1), and 1 + 3 > 3, which is the same as the first one. So all conditions are satisfied.

Alternatively, if the sides were 1, 3, and 2, that would not form a triangle because 1 + 2 = 3, which is not greater than 3. So 2 is invalid. Similarly, 4 is invalid as 1 + 3 is not greater than 4. Therefore, the only possible integer is 3.

I think that's thorough. Let me just check once more:

Given sides 1, 3, x.

Triangle inequalities:

1 + 3 > x → x < 4

1 + x > 3 → x > 2

3 + x > 1 → x > -2 (irrelevant)

So x must be between 2 and 4, exclusive. Since x is an integer, x = 3. Yep, that's correct.
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