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 possible whole-number values for side \( a \) of the triangle given sides \( b = 6 \) and \( c = 7 \), we can use the triangle inequality theorem, which states that for any triangle with sides \( a, b, \) and \( c \):

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

Let's apply these inequalities step by step:

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

2. From \( a + c > b \):
\[
a + 7 > 6 \implies a > -1 \quad \text{(This is always true since } a \text{ is positive)}
\]

3. From \( b + c > a \):
\[
6 + 7 > a \implies 13 > a \implies a < 13
\]

Combining these inequalities, we find that:
\[
1 < a < 13
\]

Since we are looking for whole-number values for \( a \), the range of possible values is:
\[
a = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
\]

Thus, the answer to complete the inequality is:

\[
a < 13
\]