Algebra
posted by Tammie on .
Just wondering if I did this correctly step by step. If not corrections are mostly appreciated!
Solve the inequality and write the solution set in interval notation. Show work/explanation.
(x+8)/(x1) >= 0
8+x/x1=1+9/x1
9/x1>=1
x1/9 < 1
x1< 9
x<8
x1>0
x>1
x+8/x> = 1
x^2+8> = x
8 > 1
x1/9<=1
x1<=9
x<=8
x1>0
x>1
Final Answer:
x < 8 OR x>1
(‡,8] ¾ (1, ‡)
Is this all of what I need to show for how I got my answer?

There are several ways to do this.
Your way is the oddest way I have ever seen. It looks like you are actually doing a long division to get
(8+x)/(x1)=1+9/(x1) (notice my brackets to make your statement from above actually true.)
Here is a logical way:
since the answer to the fraction is positive, either both top and bottom are positive or they are both negative.
That is,
[x+8≥0 and x1>0] OR [x+8≤0 and x1≤0]
[x≥8 and x>1] OR [x≤8 and x≤1]
x > 1 or x ≤ 8
so you did get the right answer.
The way I do these is this :
From the factored form, I can see two "critical values" namely 8 and 1
So my number line is split into 3 sections:
a) x ≤ 8
b) between 8 and 1
c) x >1
I then pick an arbitrary number in each region. We don't actually have to evaluate, just get the sign correctly.
a) let x=10, then / >0 , which works
b) let x=0, then +/ < 0 , does not work
c) let x=10 then +/+ > 0 , works
so x ≤ 8 or x > 1 
Thank you for the review on this topic!