Algebra

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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)/(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?

  • Algebra -

    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

  • Algebra -

    Thank you for the review on this topic!

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