Complete the inequality so that it represents the whole-number values that side a

Question

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 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:

\[ a < 13 \]