Solving Inequality: |x-1|+|x-2|>1

[solar nebula]

Homework Statement

So this is the question

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

Homework Equations

N/A

The Attempt at a Solution

I tried it, the solution seems right, but i don't know if my approach is correct.

 

[Mentallic]

Think about these absolute values like this:

At x=1 or x=2, one of the absolute values is 0, so you can call these the "roots" (thought not technically roots, I can't think of a more appropriate name for them right now), so what you want to do is check all possibilities around those roots, and the roots themselves.

Check:

x<1
For this range, both $x-1$ and $x-2$ will be negative, so the inequality you need to solve would be $$-(x-1)-(x-2)>1$$

1<x<2
Here you will have $x-1>0$ and $x-2<0$ so what you need to solve is $$(x-1)-(x-2)>1$$

x>2
For this value, both are positive so it should be clear what you need to solve here.

And then always check the "roots" themselves. Plug in the values of x=1 and x=2. By this point, you've checked all possible cases and should have your solution set.

 

[adaptation]

The way I learned it:

To solve |ax+b|>k, solve ax+b>k and ax+b<-k. (This is basically what you did.)

Then I would plug test values into the original equation to see if it makes a true or false statement. You would use the intervals (-$\infty$,1);(1,2);(2,$\infty$).

The values from those intervals that make true statements give you your solution set.

 

[Mentallic]

edit: I just realized that your question is a special case. What if it was instead $$|x-1|+|x-2|>2$$ or $$|x-1|+|x-2|>0$$ ? For the first what you need to do is check when $$|x-1|+|x-2|=2$$ and then check each interval around that.

 

[solar nebula]

Thanks Mentallic and Adaptation!