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Mathematics
OpenStudy (anonymous):

Is any taking a transition to advanced mathematics course? I need help with a proof!!!

OpenStudy (anonymous):

what kind of proof?

OpenStudy (anonymous):

I need to prove that \[|a+b| \le|a|+|b|\] and I know that it has to do with proof by cases

OpenStudy (anonymous):

ok, can see why this relationship holds before even starting?

OpenStudy (anonymous):

no..not really :( that is why I want to understand why this is the case and how I go about proving this to be true

OpenStudy (anonymous):

ok, so we have the absolute value of the sum of A and B. then the sum of the absolute value of A with the absolute value of B

OpenStudy (anonymous):

oh and we are assuming that a and b are real numbers

OpenStudy (anonymous):

sure, so any number, positive or negative, when put in the absolute value function yields a postive

OpenStudy (anonymous):

roughly speaking

OpenStudy (anonymous):

yes

OpenStudy (anonymous):

so, assume A and B are of opposite sign. if we take the absolute value of A, then abs of B, both will come out as positive numbers...this is basically the right hand side...so the right hand side is always positive, right?

OpenStudy (anonymous):

yes so that would be Case 1. right then we would have another case were both A and B are positive and A and B are both negative?

OpenStudy (anonymous):

let's still look at case 1 as you say with A and B of opposite sign....so if we sum A and B, the sum could be positive or negative, depending upon the magnitude of each....

OpenStudy (anonymous):

A+B < 0, or > 0, or = 0 (if A=(-B))

OpenStudy (anonymous):

doesn't the absolute value make the sum of A and B postive ?

OpenStudy (anonymous):

but in anycase, if of opposite sign, their sum will be less than the sum of two positive values

OpenStudy (anonymous):

we are looking at left hand side now, we know right will always be positive...but we are looking inside the absolute value function, looking right at the sum of A and B before we put that sum through the absolute value function

OpenStudy (anonymous):

haha, sorry if I confused you, does it make sense so far?

OpenStudy (anonymous):

hi sorry I couldn't post

OpenStudy (anonymous):

yeah, you can refresh the page when it gets stuck

OpenStudy (anonymous):

so I am confused as to why the right side would always be positive? can you explain that again?

OpenStudy (anonymous):

ok, how does the absolute value function work? what is the result of abs(5), what is the result of abs(-5)?

OpenStudy (anonymous):

it would be 5 in both cases. The absolute value always produces a positive answer

OpenStudy (anonymous):

right on, ok, the right side of the equation is \[\left| a \right| + \left| b \right|\]

OpenStudy (anonymous):

"The absolute value always produces a positive answer"

OpenStudy (anonymous):

"The absolute value always produces a positive answer"

OpenStudy (anonymous):

abs(a) is going to be positive, abs(b) is going to be positive as you pointed out

OpenStudy (anonymous):

the sum of two positive values is positive

OpenStudy (anonymous):

right. I think i see why the right side would always be bigger than the left, because on the right we are adding two variables after we take the absolute value of each individually whereas on the left we add them together first and then take the absolute value. Right?

OpenStudy (anonymous):

yes man...so back to the idea that A and B are of opposite sign

OpenStudy (anonymous):

yes man...so back to the idea that A and B are of opposite sign

OpenStudy (anonymous):

we know the right side will be positive, and well we know the left side will be positive because everything happens in the absolute value function

OpenStudy (anonymous):

This proof is confusing because we have to consider both A and B with positive and negative values

OpenStudy (anonymous):

right, that's the tricky part, and that's what the equation is telling you

OpenStudy (anonymous):

brb

OpenStudy (anonymous):

ok

OpenStudy (anonymous):

okay

OpenStudy (anonymous):

so can you answer it?

OpenStudy (anonymous):

case 1 = opposite sign, case 2 = same sign

OpenStudy (anonymous):

sorry again it didn't let me post

OpenStudy (anonymous):

would we have to consider greater than or equal to zero when we are doing the case where both are positive?

OpenStudy (anonymous):

and the same for the opposite sign one?

OpenStudy (anonymous):

you still there?

OpenStudy (anonymous):

when both are positive, you can ignore absolute vlaue functions right?

OpenStudy (anonymous):

it's as though that work has already been done...

OpenStudy (anonymous):

yes...but we cant when we are dealing with opposite signs

OpenStudy (anonymous):

right, when the signs are the same, it's not difficult, when signs are opposite, the sum of the values can either be positive or negative (so before applying right hand side absolute value on A + B, that sum could be greater than 0, less than 0 or 0.

OpenStudy (anonymous):

so I am confused whether that would be a single case or 3?

OpenStudy (anonymous):

so, ok, keep it as single case with 3 parts, haha..you're right, do it as 3

OpenStudy (anonymous):

couldn't we do the case for when they are both positive and both negative and then just use Without Loss of Generality? since their proofs are going to be identical?

OpenStudy (anonymous):

i guess, imagine a number-line, the both positive case takes place to the right of 0, while the both negative case takes place to the left of 0...but then the absolute value functions flip the result to be similar to that of the case of both positive

OpenStudy (anonymous):

draw the function abs(x) on a graph right now and look at the result for y...y=x, of y=f(x), where f(x) = abs(x)..i've got to go, but i can get on later

OpenStudy (anonymous):

okay thanks for your help though. I appreciate it

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