Question

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

Answer 1

Answer: Sometimes True

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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"

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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.

Answer 2

$|a-b| = |a| - |b|$ if and only if $a \ge 0\,\land\,b\ge 0$. 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:

$|x| = \left \{ {{x,\,x\ge 0} \atop {-x,\,x< 0}} \right.$, for all $x\in \mathbb{R}$ (1)

In this question we must check if $|a|-|b| = |a - b|$. Hence, we must check it for the following four cases:

(i) $a \ge 0\,\land\,b\ge 0$

(ii) $a\ge 0\,\land \,b < 0$

(iii) $a < 0\,\land\,b<0$

(iv) $a< 0\,\land b\,\ge 0$

Case I

1) $|a| \ge 0$, $|b| \ge 0$, $|a-b| \ge 0$ Definition of absolute value.

2) $a \ge 0\,\land\,b\ge 0$ Given.

3) $a - b = a - b$ Definition of absolute value/Reflexive Property

4) $|a-b| = |a| - |b|$ Result.

Case II

1) $|a| \ge 0$, $|b| \ge 0$, $|a-b| \ge 0$ Definition of absolute value.

2) $a\ge 0\,\land \,b < 0$ Given.

3) $a - b =^{?} a - (-b)$ Definition of absolute value.

4) $a - b =^{?} a + b$    $-(-x) = x$

5) $-b =^{?} b$ Compatibility with addition/Modulative property/Existence of additive inverse/Absurd found.

6) $|a-b| \ne |a| - |b|$ Reductio ad absurdum.

Case III

1) $|a| \ge 0$, $|b| \ge 0$, $|a-b| \ge 0$ Definition of absolute value.

2) $a < 0\,\land\,b<0$ Given.

3) $a-b =^{?} -a-(-b)$ Definition of absolute value.

4) $a-b =^{?} -(a -b)$   $(-1)\cdot x = -x$/Distributive property/Absurd found.

5) $|a-b| \ne |a| - |b|$ Reductio ad absurdum.

Case IV

1) $|a| \ge 0$, $|b| \ge 0$, $|a-b| \ge 0$ Definition of absolute value.

2) $a< 0\,\land b\,\ge 0$ GIven.

3) $a-b =^{?} -a - b$  Definition of absolute value.

4) $a =^{?} -a$ Compatibility with addition/Modulative property/Existence of additive inverse/Absurd found.

5) $|a-b| \ne |a| - |b|$ Reductio ad absurdum.

In a nutshell, $|a-b| = |a| - |b|$ if and only if $a \ge 0\,\land\,b\ge 0$. That is, it is sometimes the same.

Please see this question related to Absolute Values: https://brainly.com/question/11220129