Expressions without absolute value signs

In summary, there are two cases for the expression a-Abs[(a-|a-x|)] depending on the value of x: x>0 or x<0. There is only one function for this expression, which may have different parts. The expression can also be written using the Heaviside step function.
  • #1
garyljc
103
0

Homework Statement


Rewrite the following expressions without absolute value signs, treating various cases separately where neccesary


Homework Equations


a-Abs[(a-(abs)a)]


the question is do i have 2 answers to this ?
 
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  • #2
garyljc said:
the question is do i have 2 answers to this ?
Yes there will be two cases. At first inspection one might conclude that there would be four cases, but you should note that one of the absolute functions is not necessary.
 
  • #3
There is only one[/b\] function for this, which may have different parts.

If f(x)= a- |a- |x||, then, of course, you need to look at x< 0 and x> 0, then at the cases |x|> a and |x|< a.

You could also write this as a single "formula" using the Heaviside step function which is defined as H(x)= 0 if x< 0, H(x)= 1 if [itex]x\ge 0[/itex].
 
  • #4
Hootenanny said:
Yes there will be two cases. At first inspection one might conclude that there would be four cases, but you should note that one of the absolute functions is not necessary.

why would there be 4 cases ?
 
  • #5
Hmmm, the function is a-|a- |x||. If x> 0 that is a- |a- x| so if x< a, that is if a-x> 0, we have a- (a- x)= x. If x> a, that is if a- x< 0, |a-x|= -(a-x) and we have a+(a-x)= 2a- x.

If x< 0, so we have a- |a+ x|. Now if a> -x, so x+ a> 0, that is a- (a+ x)= -x. Finally if a< -x so x+ a< 0, that is a+ (a+x)= 2a+ x. Looks like 4 cases to me.
 

1. What is the purpose of using absolute value signs in expressions?

The purpose of using absolute value signs in expressions is to indicate the distance of a number from zero on a number line. It ensures that the result of the expression is always positive, regardless of the sign of the number.

2. Can an expression without absolute value signs have negative values?

Yes, an expression without absolute value signs can have negative values. The absence of absolute value signs does not restrict the expression from producing negative results.

3. How do you simplify an expression without absolute value signs?

To simplify an expression without absolute value signs, you can follow the standard order of operations (PEMDAS) and combine like terms. If there are still absolute value signs present, you can evaluate the expression for both positive and negative values and then compare the results.

4. Is it necessary to use absolute value signs in all expressions?

No, it is not necessary to use absolute value signs in all expressions. They are typically used when dealing with distance, magnitude, or absolute difference.

5. Can absolute value signs be used in complex expressions with multiple operations?

Yes, absolute value signs can be used in complex expressions with multiple operations. They follow the same rules of order of operations and can be evaluated along with other operations in the expression.

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