Just wondering if I did this correctly step by step. If not corrections are mostly appreciated!

Question

Solve the inequality and write the solution set in interval notation. Show work/explanation.

(x+8)/(x-1) >= 0

8+x/x-1=1+9/x-1

9/x-1>=-1
x-1/9 < -1
x-1< -9
x<-8
x-1>0
x>1

x+8/x> = 1

x^2+8> = x

8 > -1

x-1/9<=-1
x-1<=-9
x<=-8
x-1>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?

Answer 1

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)/(x-1)=1+9/(x-1) (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 x-1>0] OR [x+8≤0 and x-1≤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

Answer 2

Thank you for the review on this topic!