Question

The lengths of two sides of a triangle are given below. Write an inequality to represent the range of the lengths of the 3rd side.

4 and 10

Answer 1

9514 1404 393

Answer:

6 < x < 14

Explanation:

The shortest the third side can be is the difference between the given sides:

10 -4 = 6

The longest the third side can be is the sum of the given sides:

10 +4 = 14

So the inequality for side length x is ...

6 < x < 14

__

More detail

These come from the triangle inequality, which requires ...

a + b > c; b + c > a; c + a > b

Filling in 4, 10, and x, these become ...

• 4 + 10 > x ⇒ 14 > x . . . from the sum of the two given sides
• 10 + x > 4 ⇒ x > -6 . . . . . less restrictive than the next
• x + 4 > 10 ⇒ x > 6 . . . from the difference of the two given sides

In summary, 6 < x < 14.