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?
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
Thank you for the review on this topic!
To solve the inequality (x+8)/(x-1) >= 0 and write the solution in interval notation, you have provided the following steps:
1. You began by finding a common denominator for the expression on the left side of the inequality and simplified it: (x+8)/(x-1) = (1+9)/(x-1) = 9/(x-1).
2. Next, you obtained the inequality 9/(x-1) >= -1 by comparing the expression on the left side to the constant on the right side.
3. Then, you multiplied both sides of the inequality by (x-1) to eliminate the denominator, but there is an error in your work. The correct step is: x-1 < -9.
4. Afterward, you deduced that x < -8 by adding 1 to both sides of the inequality x-1 < -9. This result corresponds to the restriction x > 1, which you derived by analyzing the inequality x-1 > 0.
5. You concluded that the inequality (x+8)/x >= 1 can be written as x^2+8 >= x, and then simplified it to 8 >= -1, which is always true.
6. Finally, you identified the solution set as x < -8 or x > 1 and expressed it in interval notation as (-∞, -8] ∪ (1, ∞).
Overall, the steps you took seem quite accurate. However, there is a minor error in step 3 where you mistakenly wrote x-1 <= -9. The correct step should be x-1 < -9. Apart from that, you've shown all the necessary work and explanations to arrive at your solution.