Question

The side lengths of a triangle are 5 centimeters, 12 centimeters, and x centimeters.



Which inequality represents all of the possible values of x?

Answer options with 4 options
A.
5 is less than x which is less than 12

B.
5 is less than or equal to x which is less than or equal to 12

C.
7 is less than x which is less than 17

D.
7 is less than or equal to x which is less than or equal to 17

Answer 1

To determine the range of possible values for the side length \( x \) of a triangle with the other two side lengths being 5 cm and 12 cm, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

We will apply the triangle inequality in the following three cases:

1. \( 5 + 12 > x \)
2. \( 5 + x > 12 \)
3. \( 12 + x > 5 \)

Now, let's calculate each inequality:

1. From \( 5 + 12 > x \):
\[
17 > x \quad \text{or} \quad x < 17
\]

2. From \( 5 + x > 12 \):
\[
x > 12 - 5 \quad \text{or} \quad x > 7
\]

3. From \( 12 + x > 5 \):
\[
x > 5 - 12 \quad \text{which simplifies to} \quad x > -7 \quad \text{(which is always true because \( x \) must be positive)}
\]

The critical inequalities derived are:
- \( x > 7 \)
- \( x < 17 \)

Combining these gives us the inequality:
\[
7 < x < 17
\]

Thus, the correct answer representing all of the possible values of \( x \) is:
**C. 7 is less than x which is less than 17**.