To determine the possible whole-number lengths for side \( a \) of a triangle given the other two sides are \( b = 6 \) and \( c = 7 \), we can use the triangle inequality theorem. This theorem states that for any triangle with sides \( a \), \( b \), and \( c \), the following inequalities must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Substituting \( b = 6 \) and \( c = 7 \) into these inequalities, we get:
- \( a + 6 > 7 \)
- \( a + 7 > 6 \)
- \( 6 + 7 > a \)
Now let's solve these inequalities:
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From \( a + 6 > 7 \): \[ a > 7 - 6 \] \[ a > 1 \]
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From \( a + 7 > 6 \): \[ a > 6 - 7 \] \[ a > -1 \] (This condition is always satisfied since \( a > 1 \) is stricter.)
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From \( 6 + 7 > a \): \[ 13 > a \] \[ a < 13 \]
Putting these together, we conclude: \[ 1 < a < 13 \]
Since \( a \) must be a whole number, the possible values for \( a \) are: \[ a = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 \]
Thus, the completed inequality representing the whole-number values of side \( a \) is: \[ 2 \leq a \leq 12 \]
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