The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Given the two sides of the triangle measuring 34 and 51, we can denote the length of the third side as \( x \). According to the Triangle Inequality Theorem, we can form the following inequalities:
- \( 34 + 51 > x \)
- \( 34 + x > 51 \)
- \( 51 + x > 34 \)
Now, we will solve these inequalities step by step:
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From \( 34 + 51 > x \), we get: \[ 85 > x \quad \text{or} \quad x < 85 \]
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From \( 34 + x > 51 \), we get: \[ x > 51 - 34 \] \[ x > 17 \]
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From \( 51 + x > 34 \), we get: \[ x > 34 - 51 \] Since \( -17 \) is a negative number, this inequality does not provide any new constraint since \( x \) must be positive.
From the first two inequalities, we combine the results:
\[ 17 < x < 85 \]
Thus, the completed inequality that indicates the possible values of the third side \( x \) is:
17 < x < 85