SOLUTION: find the domain of the function: f(x)=log(base 7)(2-(x+5/x-6)i need to show the work

Question 246026: find the domain of the function: f(x)=log(base 7)(2-(x+5/x-6)i need to show the work
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

There are two issues to address here:

Denominators may not be zero.
Arguments of logarithms, no matter what the base, must not be zero or negative.

Denominator.
Let's find what x value makes the denominator zero:
x - 6 = 0
Add 6 to both sides:
x = 6
This is a value we cannot allow for x. It must be excluded from the domain.

Argument of the logarithm.
We want a positive argument:

There are different ways to solve this. One way is to subtract the terms on the left. Of course we need common denominators first:

Subtract (Be careful when subtracting!) :

This says we have a positive fraction. When do we get positive fractions? Answer: When the numerator and denominator are both positive or when they are both negative. This idea, which you probably knew well, can be expressed in the algebraic form:
( and ) or ( and )
(Look at this and see if you can understand how this says "both positive or both negative".)
Now we solve this compound inequality.
( and ) or ( and )
In the first pair, in order for x to be greater than 17 and 6, it would have to be greater than 17. In the last pair, in order for x to be less than 17 and 6, it would have to be less than 6. So
( and ) or ( and )
simplifies to
or

Since or excludes the number which makes the denominator zero, this is our domain.
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