when is the absolute value of a difference equal to the difference of the absolute values? Explain

how about

| 9 - 4| = |9| - |4|

take it from there

Oh, the absolute value and its quirky behavior! Well, let's break it down and have a little fun with it!

So, imagine you have two numbers, let's call them "x" and "y." The absolute value of a difference is just the absolute value of the result when you subtract one number from the other.

Mathematically speaking, the absolute value of (x - y) is denoted as |x - y|. On the other hand, the difference of the absolute values is |x| - |y|.

Now, when are these two things equal, you ask? Well, it turns out they're equal under one condition: whenever either (x - y) is positive or (x - y) is negative. Let's find out why!

Imagine x = 5 and y = 3. In this case, (5 - 3) equals 2. Phew, it's a positive number! Both |x - y| and |x| - |y| will be 2 in this situation.

Now, let's imagine x = 3 and y = 5. Now, (3 - 5) equals -2. Oh no, a negative number! Both |x - y| and |x| - |y| will be 2 again. Funny how it works out, right?

So, to sum it up, whenever the difference between x and y is positive or negative, the absolute value of the difference equals the difference of the absolute values. It's a clownishly simple rule that the absolute value follows!

Now, I hope I've entertained you while explaining this mathematical peculiarity. If not, well, I apologize – clown humor isn't for everyone!

The absolute value of a difference is equal to the difference of the absolute values when both values have the same sign.

To explain this, let's consider two real numbers, A and B. The absolute value of A is denoted as |A| and it represents the magnitude or distance of A from zero on the number line. The absolute value of B is denoted as |B| and represents the magnitude or distance of B from zero on the number line.

When both A and B have the same sign (either both positive or both negative), their difference can be expressed as A - B. In this case, the absolute value of their difference, which is |A - B|, will be equal to the absolute value of A minus the absolute value of B, expressed as |A| - |B|.

For example:
If A = 4 and B = 2, both numbers are positive. The difference between A and B is 4 - 2 = 2. The absolute value of their difference is |4 - 2| = |2| = 2. The difference of their absolute values is |4| - |2| = 4 - 2 = 2. So, in this case, the absolute value of the difference is equal to the difference of the absolute values.

On the other hand, if A and B have opposite signs (one positive and one negative), their difference can be expressed as A - B or B - A. In this situation, the absolute value of their difference will not be equal to the difference of the absolute values.

For example:
If A = -4 and B = 2, A is negative and B is positive. The difference between A and B can be calculated as A - B = -4 - 2 = -6. The absolute value of their difference is | -4 - 2| = |-6| = 6. The difference of their absolute values is |-4| - |2| = 4 - 2 = 2. In this case, the absolute value of the difference is not equal to the difference of the absolute values.

So, in summary, the absolute value of a difference is equal to the difference of the absolute values if and only if both values have the same sign.

To understand when the absolute value of a difference is equal to the difference of the absolute values, let's break down the mathematical expression and explain step by step.

First, let's consider two numbers, let's call them "a" and "b". The absolute value of a number is a measure of how far it is from zero on the number line, irrespective of its sign. It is denoted by vertical bars or parentheses around the number.

So, the absolute value of "a" can be represented as |a|, and the absolute value of "b" can be represented as |b|.

The expression "the absolute value of a difference" refers to the absolute value of the difference between two numbers, which is denoted as |a - b|. This means we take the difference between "a" and "b" and then take the absolute value of that difference.

On the other hand, "the difference of the absolute values" refers to the result of subtracting the absolute value of "b" from the absolute value of "a". Mathematically, it is represented as |a| - |b|.

Now, to determine when the absolute value of a difference is equal to the difference of the absolute values, we need to find under what conditions |a - b| = |a| - |b| holds true.

In general, the absolute value of a difference is not equal to the difference of the absolute values. However, there is a special case when this equality does hold true, which is when either "a" or "b" is zero.

Let's consider both scenarios:

1. When "a" is zero: If "a" is zero, then |a| is also zero. Thus, the expression becomes |0 - b| = 0 - |b|, which simplifies to |b| = -|b|. We know that the absolute value of any number is always non-negative, so -|b| is never equal to |b| for any non-zero value of "b". Therefore, in this case, the equality does not hold true.

2. When "b" is zero: Similarly, if "b" is zero, then |b| is also zero. So, the expression becomes |a - 0| = |a| - 0, which simplifies to |a| = |a|. In this case, the equality holds true since the absolute value of any number is always equal to itself.

To summarize, the absolute value of a difference is equal to the difference of the absolute values only when one of the variables ("a" or "b") is zero. In all other cases, the equality does not hold true.