Question

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

Answer 1

what kind of proof?

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

roughly speaking

Answer 9

yes

Answer 10

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?

Answer 11

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?

Answer 12

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....

Answer 13

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

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

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

Answer 18

hi
sorry I couldn't post

Answer 19

yeah, you can refresh the page when it gets stuck

Answer 20

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

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

"The absolute value always produces a positive answer"

Answer 25

"The absolute value always produces a positive answer"

Answer 26

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

Answer 27

the sum of two positive values is positive

Answer 28

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?

Answer 29

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

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

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

Answer 34

brb

Answer 35

ok

Answer 36

okay

Answer 37

so can you answer it?

Answer 38

case 1 = opposite sign, case 2 = same sign

Answer 39

sorry again it didn't let me post

Answer 40

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

Answer 41

and the same for the opposite sign one?

Answer 42

you still there?

Answer 43

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

Answer 44

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

Answer 45

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

Answer 46

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.

Answer 47

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

Answer 48

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

Answer 49

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?

Answer 50

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

Answer 51

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

Answer 52

okay thanks for your help though. I appreciate it