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anonymous

  • 5 years ago

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

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  1. anonymous
    • 5 years ago
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    what kind of proof?

  2. anonymous
    • 5 years ago
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    I need to prove that \[|a+b| \le|a|+|b|\] and I know that it has to do with proof by cases

  3. anonymous
    • 5 years ago
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    ok, can see why this relationship holds before even starting?

  4. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    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
    • 5 years ago
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    oh and we are assuming that a and b are real numbers

  7. anonymous
    • 5 years ago
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    sure, so any number, positive or negative, when put in the absolute value function yields a postive

  8. anonymous
    • 5 years ago
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    roughly speaking

  9. anonymous
    • 5 years ago
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    yes

  10. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    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
    • 5 years ago
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    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
    • 5 years ago
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    A+B < 0, or > 0, or = 0 (if A=(-B))

  14. anonymous
    • 5 years ago
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    doesn't the absolute value make the sum of A and B postive ?

  15. anonymous
    • 5 years ago
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    but in anycase, if of opposite sign, their sum will be less than the sum of two positive values

  16. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    haha, sorry if I confused you, does it make sense so far?

  18. anonymous
    • 5 years ago
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    hi sorry I couldn't post

  19. anonymous
    • 5 years ago
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    yeah, you can refresh the page when it gets stuck

  20. anonymous
    • 5 years ago
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    so I am confused as to why the right side would always be positive? can you explain that again?

  21. anonymous
    • 5 years ago
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    ok, how does the absolute value function work? what is the result of abs(5), what is the result of abs(-5)?

  22. anonymous
    • 5 years ago
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    it would be 5 in both cases. The absolute value always produces a positive answer

  23. anonymous
    • 5 years ago
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    right on, ok, the right side of the equation is \[\left| a \right| + \left| b \right|\]

  24. anonymous
    • 5 years ago
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    "The absolute value always produces a positive answer"

  25. anonymous
    • 5 years ago
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    "The absolute value always produces a positive answer"

  26. anonymous
    • 5 years ago
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    abs(a) is going to be positive, abs(b) is going to be positive as you pointed out

  27. anonymous
    • 5 years ago
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    the sum of two positive values is positive

  28. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    yes man...so back to the idea that A and B are of opposite sign

  30. anonymous
    • 5 years ago
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    yes man...so back to the idea that A and B are of opposite sign

  31. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    This proof is confusing because we have to consider both A and B with positive and negative values

  33. anonymous
    • 5 years ago
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    right, that's the tricky part, and that's what the equation is telling you

  34. anonymous
    • 5 years ago
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    brb

  35. anonymous
    • 5 years ago
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    ok

  36. anonymous
    • 5 years ago
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    okay

  37. anonymous
    • 5 years ago
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    so can you answer it?

  38. anonymous
    • 5 years ago
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    case 1 = opposite sign, case 2 = same sign

  39. anonymous
    • 5 years ago
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    sorry again it didn't let me post

  40. anonymous
    • 5 years ago
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    would we have to consider greater than or equal to zero when we are doing the case where both are positive?

  41. anonymous
    • 5 years ago
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    and the same for the opposite sign one?

  42. anonymous
    • 5 years ago
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    you still there?

  43. anonymous
    • 5 years ago
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    when both are positive, you can ignore absolute vlaue functions right?

  44. anonymous
    • 5 years ago
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    it's as though that work has already been done...

  45. anonymous
    • 5 years ago
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    yes...but we cant when we are dealing with opposite signs

  46. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    so I am confused whether that would be a single case or 3?

  48. anonymous
    • 5 years ago
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    so, ok, keep it as single case with 3 parts, haha..you're right, do it as 3

  49. anonymous
    • 5 years ago
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    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
    • 5 years ago
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    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
    • 5 years ago
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    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
    • 5 years ago
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    okay thanks for your help though. I appreciate it

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