Meaning of absolute value |x|?

Question

Understanding: It actually means the distance of x from the origin.

My questions is why is |x+A| equals the distance of x from -A, or why is |x-A| equals the distance of x from +A. Please help me to understand the actual meaning here?

BrushMyQuant · Answer

Let's take two cases to understand what is | X - A | and | X + A |

1. | X - A | : Both X and A are positive
X - A will be nothing but the difference between X and A, which can have negative or positive sign depending on the values of X and A
But, | X - A | will always be the positive difference between values of X and A, which is nothing but the distance between X and A

2. | X + A | : Both X and A are positive 
 X + A will be nothing but the summation of values of X and A
So, Distance of X from origin (which is nothing but X) + Distance of A from origin (which is nothing but A)
= Distance of X from origin (which is nothing but X) + Distance of -A from origin (distance of A and -A from origin both are same)
= Distance between X and -A
| X + A | = | X - (-A) | => Distance between X and -A

Hope it helps!

souvonik2k · Answer

Hi
|x| means distance between x and 0 or |x-0|.
Since distance can never be zero, |x-0| is always positive.
Same logic applies for |x-a| or |x+a|(same as |x-(-a)|)

Please give kudos , if u find this post useful.

ccooley · Answer

I think this is easier to understand when you look at x-A, rather than x+A, so let's start there.

If x - A is positive already, then the value of |x - A| is just x - A itself.

Let's relate that to distances on the number line: if x - A is positive, that means x is bigger than A. So, x is to the right of A on the number line. You don't necessarily know whether x and A themselves are positive or negative. But you do know that x is to the right of A, wherever they are.

https://i.imgur.com/OECIw5F.png

In this scenario, the distance between x and A is equal to x - A. That's true even if A is negative. Here's why:

https://i.imgur.com/gwXhgJI.png

So, if x-A is positive, the distance between x and A is just x-A. Since x-A is positive, x-A = |x-A|. (Taking the absolute value of a positive number doesn't change it.)

If x-A is negative, that means x is to the left of A. Therefore, the distance between them is A - x, or -(x - A). (Notice how you always subtract the number on the left on the number line, from the number on the right.) Since x - A is negative, when we take its absolute value, we 'flip its sign' to make it positive: |x - A| = -(x - A).

Therefore, the distance is equal to the absolute value in this case too.

Meaning of absolute value |x|?

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