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?
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).
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
Opps. I've corrected the mistake. has been changed to: since ±2 is positive 2 AND negative 2. Hmm... interesting perspective. I suppose it might be a syntax issue. 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.
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...., right again. 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.
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.
Can I give you some advice? Actually, I'm going to do it anyway 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 :-)