Determine whether the absolute value of a difference is sometimes, always, or never the same as the difference of the absolute values. Ia-bI = IaI - IbI

if and only if . That is, it is sometimes the same.

Absolute Value is a mathematic Form which represents either the Magnitude or the Distance respect to Origin of a given Real Number, whose is depicted below:

, for all (1)

In this question we must check if . Hence, we must check it for the following four cases:

(i)

(ii)

(iii)

(iv)

Case I

1) , , Definition of absolute value.

2) Given.

3) Definition of absolute value/Reflexive Property

4) Result.

Case II

1) , , Definition of absolute value.

2) Given.

3) Definition of absolute value.

4)    

5) Compatibility with addition/Modulative property/Existence of additive inverse/Absurd found.

6) Reductio ad absurdum.

Case III

1) , , Definition of absolute value.

2) Given.

3) Definition of absolute value.

4)   /Distributive property/Absurd found.

5) Reductio ad absurdum.

Case IV

1) , , Definition of absolute value.

2) GIven.

3)  Definition of absolute value.

4) Compatibility with addition/Modulative property/Existence of additive inverse/Absurd found.

5) Reductio ad absurdum.

In a nutshell, if and only if . That is, it is sometimes the same.

Please see this question related to Absolute Values: link

Explanation:

Let's see what happens when a = b

Let's pick something like a = 2, so b = 2.

|a-b| = |2-2| = |0| = 0

|a|-|b| = |2|-|2| = 2-2 = 0

Both result in 0.

We see that |a-b| = |a|-|b| is true for this example.

The given equation is either "sometimes true" or "always true"

Now let's pick something where a < b. Let a = 1 and b = 2

|a-b| = |1-2| = |-1| = 1

|a|-|b| = |1|-|2| = 1-2 = -1

We get different results, so |a-b| = |a|-|b| is not true for this example.

So this leads us to say the original equation is sometimes true.

4444444444444 i think