This is a little embarassing, but it seems like I do not understand why the answers are what they are for these inequalities.
First, I factor the left-hand side.
So, if we divide both sides by '3x+1'.
Here's the part I don't understand. Naturally, I think that if we divide 'x-1' from both sides. We would get
But this can not be right. By counterexample, let x = 0; this inequality is false.
It should be .
I believe the reason for this is since for (x-1), if x is a negative number, then (x-1) < 0. So in order for the inequality to be equal then '3x+1' must also be negative (since a negative times a negative equals a positive) which explains why it should be
Hence we solve, we should get . What am I doing wrong at (*)?
2. Another confusing inequality.
I multiply both sides by '3-x'
This answer makes no sense. If x = 4, then the inequality is false.
Then, I tried a different way. I add both sides by 2
So, . Since, x can not be 3, or else the denominator is zero. I conclude that
What am I doing wrong at (*)?