Answer:
[tex]3\leq x\leq 6[/tex]
Step-by-step explanation:
So we want to find the domain of:
[tex]\sqrt{6-x}-\sqrt{3x-9}[/tex]
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:
[tex]6-x\geq 0[/tex]
Add x to both sides:
[tex]6\geq x[/tex]
Flip:
[tex]x\leq 6[/tex]
So, for the first radical, x must be less than or equal to 6.
Second radical:
[tex]3x-9\geq 0[/tex]
Add 9 to both sides:
[tex]3x\geq 9[/tex]
Divide both sides by 3:
[tex]x\geq 3[/tex]
So, our domain is:
[tex]x\leq 6\text{ and } x\geq 3[/tex]
Therefore, as a compound inequality, this is:
[tex]3\leq x\leq 6[/tex]
This is our domain.
And we're done!
