Is the following the only reason why |x| ≠ ±x?

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  • #1
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Assumption: |x| is unconditionally equal to ±x.

This makes sense because if you take a look at a graph of y=|x|, and plot any horizontal line y=C where C is some constant, you will always have two solutions: one is positive and one is negative.

But if we substitute any number into x, then we realize that this actually contradicts:

|x| = ±x
Let x = 2
|2| = ±2
2 = ±2
2 = 2 OR 2 = -2

Am I missing something or is the only reason why they aren't unconditionally equal?
 
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Answers and Replies

  • #2
CompuChip
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That final statement is true, isn't it? So I don't see an issue there.

I think the problem you are running into is that "±x" isn't well-defined notation, whereas |x| is unambiguously defined. People often use it as shorthand, as you have done, for example in statements like "The solution of x² = 4 is x = ±2", but that is just an informal way of saying "The solutions of x² = 4 are x = -2 and x = +2".

You could write "The solution of x² = 4 is |x| = 2" but that is technically something different - what you are saying then is: "The solutions to the equation x² = 4 are the same as the solutions to the equation |x| = 2" (and the solutions to both equations are x = 2 and x = -2).
 
  • #3
the modulus of x equals x if x is non-negative and -x if x is less than zero. It does not equal ±x.

The equation mod(x)=2 has the solutions x= ± 2
 
  • #4
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That final statement is true, isn't it? So I don't see an issue there.
Opps. I've corrected the mistake.

2 = 2 OR 2 = -2
has been changed to:

2 = 2 AND 2 = -2
since ±2 is positive 2 AND negative 2.

I think the problem you are running into is that "±x" isn't well-defined notation, whereas |x| is unambiguously defined. People often use it as shorthand, as you have done, for example in statements like "The solution of x² = 4 is x = ±2", but that is just an informal way of saying "The solutions of x² = 4 are x = -2 and x = +2".

You could write "The solution of x² = 4 is |x| = 2" but that is technically something different - what you are saying then is: "The solutions to the equation x² = 4 are the same as the solutions to the equation |x| = 2" (and the solutions to both equations are x = 2 and x = -2).
Hmm... interesting perspective. I suppose it might be a syntax issue.

the modulus of x equals x if x is non-negative and -x if x is less than zero. It does not equal ±x.

The equation mod(x)=2 has the solutions x= ± 2
You mean absolute value function instead of modulus function right?

The issue is that you don't know if x is negative or non-negative.
 
  • #5
CompuChip
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since ±2 is positive 2 AND negative 2.
No, it can only have one value. "x = 2 and x = -2" does not make sense, as a variable can only have one value at the time.
As I said, it is usually used as shorthand for "+2 or -2".
 
  • #6
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No, it can only have one value. "x = 2 and x = -2" does not make sense, as a variable can only have one value at the time.
As I said, it is usually used as shorthand for "+2 or -2".
Hmm... you're right. I changed it back.
 
  • #7
CompuChip
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You mean absolute value function instead of modulus function right?

The issue is that you don't know if x is negative or non-negative.
That is the same with x in ##x^2=4##. Is it an issue for you there?
 
  • #8
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That is the same with x in ##x^2=4##. Is it an issue for you there?
Hmm...., right again.

the modulus of x equals x if x is non-negative and -x if x is less than zero. It does not equal ±x.
How does it not equal ±x? It's equal to +x or -x depending on whether x is non-negative or negative.
 
  • #9
D H
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How does it not equal ±x? It's equal to +x or -x depending on whether x is non-negative or negative.
Because ±x is a multivalued function of x with two branches while |x| is a true function of x. Note that the term "multivalued function" is a bit of a misnomer. A multivalued function is not a function.
 
  • #10
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Because ±x is a multivalued function of x with two branches while |x| is a true function of x. Note that the term "multivalued function" is a bit of a misnomer. A multivalued function is not a function.
So the difference is that |x| has the condition and gives you the right solution depending on the condition and ±x just says either +x OR -x but it doesn't give you the condition?
 
  • #11
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So the difference is that |x| has the condition and gives you the right solution depending on the condition and ±x just says either +x OR -x but it doesn't give you the condition?
No, the difference is that |x| represents a single number. ±x represents two numbers, as long as x isn't 0.
 
  • #12
CompuChip
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Can I give you some advice?

Actually, I'm going to do it anyway :-p

As long as you don't completely understand "±x", avoid using it. As I pointed out before, it does not have any formal definition like |x| does - it is merely used as shorthand. For the time being, I would suggest that you focus on getting the basics right. Writing "x = -2 V x = 2" is hardly more work than "x = ±2", it is unambiguous and it doesn't confuse anyone, including yourself.

Once you have properly learned about functions and branch cuts you may be more sloppy :-)
 

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