We are asked to find when the of a difference of two numbers is equal to the difference of their absolute values. Let's write this so that we can better visualize the meaning of the statement.
| a -b | = | a | - | b |
If we try to analyze the statement, we will see that it can get confusing due to the many possibilities. One way to solve this is to organize all these different cases using example values for each case.
Condition | a | > | b |
First, we will consider that the absolute value of the first number is greater than the absolute value of the second. Under this condition, we can still have different cases: both numbers are , both are , or one is positive and one is negative. Let's work out the case when both are negative using the example values a=-8 and b=-3.
| a -b |
| -8 -( -3) |
| -8 +3 |
| -5 |
5
Now let's work out the right-hand side of the statement.
| a | - | b |
| -8 | - | -3 |
8 - 3
5
We will continue to do this for all the possible different cases.
| Case
|
| a -b |
|
| a | - | b |
|
| a -b | ? = | a | - | b |
|
| a, b <0
|
| -8 - (-3) | = 5
|
| -8 | - | (-3) | = 5
|
✓
|
| a>0, b <0
|
| 8 - (-3) | = 11
|
| 8 | - | (-3) | = 5
|
*
|
| a, b >0
|
| 8 - 3 | = 5
|
| 8 | - | 3 | = 5
|
✓
|
| a<0, b >0
|
| -8 - 3 | = 11
|
| -8 | - | 3 | = 5
|
*
|
Condition | b | > | a |
Now, let's consider the possibilities when | b | > | a |.
| Case
|
| a -b |
|
| a | - | b |
|
| a -b | ? = | a | - | b |
|
| a, b <0
|
| -3 - (-8)| = 5
|
| -3|- | (-8)| = -5
|
*
|
| a>0, b <0
|
|3 - (-8) | = 11
|
| 3 | - | (-8) | = -5
|
*
|
| a, b >0
|
|3 - 8 | = 5
|
| 3 | - |8 | = -5
|
*
|
| a<0, b >0
|
| -3 - 8 | = 11
|
| -3 | - | 8 | = -5
|
*
|
We can see that there is no possible case that makes the statement true when | b | > | a |.
Condition | a | = | b |
Finally let's consider | a | = | b |.
| Case
|
| a -b |
|
| a | - | b |
|
| a -b | ? = | a | - | b |
|
| a, b <0
|
| -3 - (-3) | = 0
|
| -3 | - | (-3) | = 0
|
✓
|
| a>0, b <0
|
| 3 - (-3) | = 6
|
| 3 | - | (-3) | = 0
|
*
|
| a, b >0
|
| 3 - 3 | = 0
|
| 3 | - | 3 | = 0
|
✓
|
| a<0, b >0
|
| -3 - 3 | = 6
|
| -3 | - | 3 | = 0
|
*
|
Conclusions
Looking at the results from the previous sections we can see that the statement is true only when
- | a | > | b | and both numbers have the same sign.
- | a | = | b | and both numbers have the same sign.
We can summarize these results and say that the statement is true when | a | ≥ | b | and both numbers have the same sign.
