MHB Is the Definition of Absolute Value Always True?

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The definition of absolute value states that |x| equals x when x is greater than or equal to 0, and |x| equals -x when x is less than 0. This allows for rewriting expressions without absolute values, such as |x - 3|. When x is less than 3, |x - 3| can be expressed as 3 - x, confirming that |2 - 3| equals 1, which is consistent with this definition. Conversely, for x greater than or equal to 3, |x - 3| simplifies to x - 3. The absolute value definition holds true across these scenarios.
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The definition of absolute value states the following:

| x | = x when x is > or = 0

| x | = -x when x < 0

I can use the above definition to rewrite expressions that do not contain absolute values.

| x - 3 |, where x < 3

If x < 3, then we can say that (x - 3) is less than 0.

So, -(x - 3) = -x + 3 = (3 - x).

Correct?
 
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Yes, for x< 3, |x- 3|= 3- x. For example, if x= 2, |2- 3|= |-1|= 1= 3- 2.

And, of course, if [math]x\ge 3[/math], |x- 3|= x- 3.
 
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