## anonymous 5 years ago Is any taking a transition to advanced mathematics course? I need help with a proof!!!

1. anonymous

what kind of proof?

2. anonymous

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

3. anonymous

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

4. 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

5. 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

6. anonymous

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

7. anonymous

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

8. anonymous

roughly speaking

9. anonymous

yes

10. 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?

11. 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?

12. 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....

13. anonymous

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

14. anonymous

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

15. anonymous

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

16. 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

17. anonymous

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

18. anonymous

hi sorry I couldn't post

19. anonymous

yeah, you can refresh the page when it gets stuck

20. anonymous

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

21. anonymous

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

22. anonymous

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

23. anonymous

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

24. anonymous

"The absolute value always produces a positive answer"

25. anonymous

"The absolute value always produces a positive answer"

26. anonymous

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

27. anonymous

the sum of two positive values is positive

28. 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?

29. anonymous

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

30. anonymous

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

31. 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

32. anonymous

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

33. anonymous

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

34. anonymous

brb

35. anonymous

ok

36. anonymous

okay

37. anonymous

38. anonymous

case 1 = opposite sign, case 2 = same sign

39. anonymous

sorry again it didn't let me post

40. anonymous

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

41. anonymous

and the same for the opposite sign one?

42. anonymous

you still there?

43. anonymous

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

44. anonymous

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

45. anonymous

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

46. 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.

47. anonymous

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

48. anonymous

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

49. 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?

50. 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

51. 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

52. anonymous

okay thanks for your help though. I appreciate it