Question

Geometry (16)

The lengths of the sides of a sandbox are 9ft, 6ft and n ft. Write a compound inequality that describes all possible lengths of n.

Answer 1

a = 9 and b = 6 are the two given sides

The third missing side n can be described through the following inequality
a-b < n < a+b
where a > b ('a' is larger than b)
This is the triangle inequality theorem

So,
a-b < n < a+b
9-6 < n < 9+6
3 < n < 15

Answer: 3 < n < 15
(which means you can pick any number from 3 to 15 for n. You cannot pick 3. You cannot pick 15)

Answer 2

Answer: The required compound inequality is
`3<n<15.`

Step-by-step explanation: We are given that the lengths of the sides of a sandbox are 9 ft, 6 ft and n ft.

We are to write a compound inequality that describes all possible lengths of n.

Since the sandbox has three sides, so it must be a triangle.

Also, we know that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

So, we have

`9+6>n\\\\\Rightarrow 15>n\\\\\Rightarrow n<15~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)`

`6+n>9\\\\\Rightarrow n>9-6\\\\\Rightarrow n>3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)`

and

`9+n>6\\\\\Rightarrow n>6-9\\\\\Rightarrow n>-3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(iii)`

Combining inequalities (i), (ii) and (iii), we get

`3<n<15.`

Thus, the required compound inequality is
`3<n<15.`