Understanding Absolute Values: Tips and Tricks to Mastering the Concept

[JasonRox]

This is easy, but for some reason I can't grasp the idea.

|x-1|+|x-2|>1

I know it means that the distance between x and 2, and x and 1 is larger than 1.

It isn't school related, and I did search online, but they are simple ones like...

|x-1|>1

Can anyone help me?

Thanks.

 

[Päällikkö]

I'll use the simpler one as an example:
|x-1| = \left\{ \begin{array}{l}<br /> x-1, \mbox{when } x \ge 1 \\<br /> -(x-1), \mbox{when } x &lt; 1 \\<br /> \end{array} \right.

The equation in parts:
x \ge 1:
x-1&gt;1
x&gt;2

x &lt; 1:
-(x-1)&gt;1
-x+1&gt;1
x&lt;0

Combining the answers we get:
x&lt;0 \vee x&gt;2

This said, can you solve the more complex one?

 

[arildno]

Construct 3 systems of inequalities the union of whose solutions will be the solutions of your original inequality:

1. |x-1|+|x-2|&gt;1, x&gt;{2}
Using the info from the second inequality to simplify the first yields:
x-1+x-2>2\to{x}>2
This system of inequalities is fulfilled for x&gt;{2}

2. |x-1|+|x-2|&gt;1, 1\leq{x}\leq{2}
Using the second inequality to simplify the first:
x-1+2-x&gt;1\to{1}&gt;1
That is, no solutions exist.

3. |x-1|+|x-2|&gt;1,x\leq{1}
Using info from the second inequality to simplify the first, we get:
3-2x&gt;1 that is, x<1, along with the inequality x&lt;{1}
This means that x<1 is the solutions in this subdomain.

Thus, x<1 or x>2 are solutions to your original inequality.

 

[JasonRox]

So, basically if I get something along the lines of...

|x-a|+|x-b|>c

I should just have 3 cases.

Thanks, arildno.

I might be back with some more.