Use the image to answer the question.

Question

Complete the inequality so that it represents the whole-number values that side a
could be to create a triangle.
An illustration of a triangle shows an equation along the base as b equals 6 and the hypotenuse as c equals 7. The third side on the triangle is labeled as a.

(1 point)
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Answer 1

To determine the range of whole-number values for side \( a \) that would allow for the formation of a triangle with the given sides \( b = 6 \) and \( c = 7 \), we can use the triangle inequality theorem. This theorem states that for any triangle with sides \( a \), \( b \), and \( c \):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

Substituting the known values for \( b \) and \( c \):

1. \( a + 6 > 7 \) implies \( a > 1 \)
2. \( a + 7 > 6 \) implies \( a > -1 \) (this is always true for positive \( a \))
3. \( 6 + 7 > a \) implies \( a < 13 \)

From the first and third inequalities, we can combine them to find:

\[ 1 < a < 13 \]

Thus, the complete inequality representing the whole-number values that side \( a \) could be to create a triangle is:

1 < a < 13