Equations with multiple absolute value operators

[pinsky]

Hello there. I'm having some problems with absolute values when they contain multiple "abs" operators and some other numbers outside the "abs"-es.

For example:

$$\left | x+2 \right | - \left | x \right | > 1$$

If i check it for the positive scenario, the result is true for all x-es.

$$x+2 - x > 1$$

I can't conceptually see how to arrange the equations for the second scenario.

I've tried the following

$$-x-2 < 1 + x$$

but that didn't give the right result. The correct result is -1/2<x

 

[Mark44]

Hello there. I'm having some problems with absolute values when they contain multiple "abs" operators and some other numbers outside the "abs"-es.

For example:

$$\left | x+2 \right | - \left | x \right | > 1$$

If i check it for the positive scenario, the result is true for all x-es.

$$x+2 - x > 1$$

I can't conceptually see how to arrange the equations for the second scenario.

I've tried the following

$$-x-2 < 1 + x$$

but that didn't give the right result. The correct result is -1/2<x

I would write the inequality as |x + 2| > |x| + 1.

You can get rid of the absolute values by looking at three separate intervals: -2 < x, -2 < x < 0, and x > 0. Each of these intervals defines one case. The final answer will be the union of the solution sets for the three cases.

 

[pinsky]

So is that the general way of doing it?

It seams a bit like a non analytical method :P

 

[Mark44]

This technique uses the definition of the absolute value, so there's nothing non-analytical about it.

 

[pinsky]

Very well.

Thank you for helping.

SOLVED