Question

Use the image to answer the 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:
\[
2 \leq a < 13
\]