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)
<a<

Answer 1

To determine the range of whole-number values that side \( a \) could take in a triangle given the lengths of 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 into these inequalities:

1. From \( a + b > c \): \[ a + 6 > 7 \ a > 1 \]

2. From \( a + c > b \): \[ a + 7 > 6 \ a > -1 \] (This inequality does not provide new information since \( a \) must be a whole number greater than 1.)

3. From \( b + c > a \): \[ 6 + 7 > a \ 13 > a \quad \text{or} \quad a < 13 \]

Combining the results from the inequalities: \[ 1 < a < 13 \]

Since \( a \) must be a whole number, the complete inequality representing the possible values for \( a \) is: \[ 2 \leq a \leq 12 \]

Therefore, the final answer to complete the inequality is: \[ a < 13 \]

Or more specifically, since you need the end of the inequality expressed simply, you can write: \[ 2 \leq a < 13 \]