This is a little embarassing, but it seems like I do not understand why the answers are what they are for these inequalities.

1.

First, I factor the left-hand side.

So, if we divide both sides by '3x+1'.

Thus,

Here's the part I don't understand. Naturally, I think that if we divide 'x-1' from both sides. We would get

which yields

(*)

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

and not

Hence we solve, we should get . What am I doing wrong at (*)?

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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 (*)?

Thank you.