Question

Domain of root(6-x)-root(3x-9)

Answer 1

`3\leq x\leq 6`

Explanation:

So we want to find the domain of:

`√(6-x)-√(3x-9)`

Recall that the radicands cannot be negative. In other words, they must be greater than or equal to 0. So, to solve the domain, determine the restrictions of each radical:

`6-x\geq 0`

Add x to both sides:

`6\geq x`

Flip:

`x\leq 6`

So, for the first radical, x must be less than or equal to 6.

Second radical:

`3x-9\geq 0`

Add 9 to both sides:

`3x\geq 9`

Divide both sides by 3:

`x\geq 3`

So, our domain is:

`x\leq 6\text{ and } x\geq 3`

Therefore, as a compound inequality, this is:

`3\leq x\leq 6`

This is our domain.

And we're done!