To find the range of possible lengths for the third side \( x \) of a triangle with two sides of lengths 5 and 8, we can use the triangle inequality theorem. The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side.
We can set up the inequalities as follows:
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The sum of the lengths of the two known sides must be greater than the length of the third side: \[ 5 + 8 > x \ 13 > x \ x < 13 \]
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The sum of one known side and the third side must be greater than the other known side: \[ 5 + x > 8 \ x > 3 \]
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The sum of the other known side and the third side must also be greater than the first known side: \[ 8 + x > 5 \ x > -3 \]
The most restrictive conditions from these inequalities are \( x < 13 \) and \( x > 3 \).
Thus, the range of possible lengths for the third side \( x \) is: \[ 3 < x < 13 \]
The numerical values in order are:
- Lower bound: \( 3 \)
- Upper bound: \( 13 \)
So, the answer is:
- First box: \( 3 \)
- Second box: \( 13 \)
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