When is the absolute value of a difference equal to the difference equal to the difference of the absolute values? Explain
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To understand when the absolute value of a difference is equal to the difference of the absolute values, we can break it down into two cases and analyze each one separately:
Case 1: When both values are positive.
Let's consider two positive numbers, a and b. The absolute value of their difference, |a - b|, will always be equal to the difference of their absolute values, |a| - |b|. This is because both a and b are positive, so their absolute values are the same as the original values. The subtraction of |a| - |b| simply removes the common positive sign, resulting in the same value as the absolute difference.
Example:
a = 5, b = 3
|a - b| = |5 - 3| = |2| = 2
|a| - |b| = |5| - |3| = 5 - 3 = 2
In this case, the absolute value of the difference indeed equals the difference of the absolute values.
Case 2: When either value is negative or zero.
Now, if we have one or both negative numbers, or if one or both numbers are zero, the equality |a - b| = |a| - |b| does NOT hold. This is because the absolute value of a negative number is its negation, and the absolute value of zero is simply zero. So, subtracting the absolute values in these cases would introduce an incorrect sign.
Example:
a = -2, b = 4
|a - b| = |-2 - 4| = |-6| = 6
|a| - |b| = |-2| - |4| = 2 - 4 = -2
Here, the absolute value of the difference (6) does not equal the difference of the absolute values (-2). The negative sign in the expression |a| - |b| causes the inequality.
Therefore, the absolute value of a difference is equal to the difference of the absolute values ONLY when both values are positive. In all other cases, the equality does not hold.