A triangle has two sides of lengths 5 and 8. What is the range of possible lengths for the third side lengths, which is represented by x? Type the numerical values in the boxes and remember to put them in the correct order.

Question

___<x<____

Answer 1

To find the range of possible lengths for the third side \( x \) of a triangle when the two other sides are of lengths 5 and 8, we can use the triangle inequality theorem.

The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. This gives us the following conditions:

1. \( 5 + 8 > x \)
2. \( 5 + x > 8 \)
3. \( 8 + x > 5 \)

Let's simplify each of these inequalities:

1. From \( 5 + 8 > x \): \[ 13 > x \quad \text{or} \quad x < 13 \]

2. From \( 5 + x > 8 \): \[ x > 8 - 5 \quad \Rightarrow \quad x > 3 \]

3. From \( 8 + x > 5 \): \[ x > 5 - 8 \quad \Rightarrow \quad x > -3 \] (This condition is always satisfied since \( x > 3 \) is stronger.)

Combining the results of the inequalities, we have: \[ 3 < x < 13 \]

Thus, the range of possible lengths for the third side \( x \) is: \[ 3 < x < 13 \]

In numerical values for the boxes, you would write: 3 (lower bound) and 13 (upper bound).