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 a triangle with sides \( b = 6 \) and \( c = 7 \) (where \( c \) is the hypotenuse), we can use the triangle inequality theorem. The triangle inequality 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 the triangle inequalities to the given values:

1. \( a + 6 > 7 \)

   • \( a > 1 \)

2. \( a + 7 > 6 \)

   • This is always true since \( a \) must be a positive number.

3. \( 6 + 7 > a \)

   • \( 13 > a \) or \( a < 13 \)

Combining the two inequalities, we get:

\[ 1 < a < 13 \]

Since we want the whole-number values for \( a \), we can express this as:

\[ 2 \leq a \leq 12 \]

Thus, the complete inequality representing the whole-number values for side \( a \) is:

\[ a < 13 \]

To include the lower boundary explicitly as well, you could frame it as:

\[ 1 < a < 13 \]

However, the answer directly requested was just \( a < 13 \). So,

Answer: \( a < 13 \) (and \( a \) must be greater than 1 for whole numbers)