Are |x+|y+z|| and ||x+y|+z| equal algebraically?

[marksyncm]

Hello,

How does one go about algebraically checking if $|x+|y+z||$ and $||x+y|+z|$ are equal?

 

[fresh_42]

Hello,

How does one go about algebraically checking if $|x+|y+z||$ and $||x+y|+z|$ are equal?

You mean as in $|x+|y+z||=|0+|-1+1||=0 \neq 2= ||0-1|+1|=||x+y|+z|\,$?

 

[andrewkirk]

The way that always works is to consider all possible cases. There are four absolute value signs, for each of which there are two possibilities: that the number they contain is negative or not. That gives $2^4=16$ cases to consider. If you consider each one in turn, and show that in that case the equality holds, and that is true for all cases, you will have proven the whole thing. If even one fails, it is disproven.

Often there will be a quicker way, specific to the particularities of the problem, that uses things like the triangle inequality. But if you can't find one, you can always fall back on the above 'brute force' method.

 

[fresh_42]

Another possibility is to consider the case of one variable zero, e.g. $x=0$. If it holds you can assume the case $x\neq 0$ and divide the entire equation by $|x|$. Thus you will have only two variables and $\pm 1$ left.

 

[marksyncm]

andrewkirk, this is exactly what I was looking for. Thank you.

fresh_42, thank you for your input as well, your last post is an interesting way of approaching this.