# Absolute value functions.

I've been really struggling with absolute value functions for a while now I'm hoping to get some help so that I can finally understand them.

How would you go about solving this absolute value inequality? I'll show my attempt to solve them and then hopefully you can tell me where I am going wrong.

$\left | 2x-1 \right |≤ \left | 3x \right |$

So next I would find the different cases.

$2x-1<0$ or $x<\frac{1}{2}$ case 1
$2x-1≥0$ or $x≥\frac{1}{2}$  case 2

and then solve

$2x-1≤3x$  = $-1≤x$
or
$2x-1≤3x=-x≤1=x≤-1$

My first problem arises here I'm not sure which way I am supposed to solve it, because one way I get x ≤-1 and another way I get -1≤x. What am I doing wrong here? The next problem arises which case do I check to make sure both conditions are right for instance do I do

$x<\frac{1}{2}$ and $x≤-1$         or          $x≥\frac{1}{2}$ and $x≤-1$
Because in the first case, both conditions are met but in the second case both conditions are not met.

Now if I try and solve case two I have the same problems.

$2x-1≥-3x = 5x≥1 = x ≥\frac{1}{5}$
or
$2x-1≥-3x=-1≥-5x=\frac{1}{5} ≥x$

Now again I have the same problem where I have two values for x and again which case do a check to see if both conditions are met. This is the problem I am struggling with and I am not sure what I am doing wrong.

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• I understand the logic behind the first way of solving it, but the second way I'm a little unsure when it comes to the intervals I believe they're called. since the bounty was only $10 I don't expect you to do an in-depth explanation of them but if I open up another question on the website and set the bounty to$20 will you be able to answer my further questions? What time will you be online? I really like your way of explaining and I don't want someone else taking the bounty.
OK, in a couple of hours when I get home, I should be able to provide more explanation on selecting the intervals. You can pay me here, by adding $20 or more tip. Sound good? • Added detailed steps. With these steps, you should hopefully be all set. When tipping, please remember 20% is taken by the website. Thanks. The answer is accepted. Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer. • answered • 1687 views •$10.00