Hello,
Can someone help me with my math problem?
Does |k – n| = |k| – |n| for all integers k and n? If so, give 3 examples.If not, give a counterexample.
Of course, I'll be happy to help you with your math problem!
To determine whether |k - n| = |k| - |n| holds true for all integers k and n, we can start by considering the definition of absolute value.
The absolute value of a number x, denoted as |x|, is a measure of the distance between x and 0 on the number line. By definition, |x| is always positive or zero.
Now let's apply this to our equation: |k - n| = |k| - |n|
1. If k and n are both positive integers:
Let's say k = 5 and n = 3.
|k - n| = |5 - 3| = |2| = 2
|k| - |n| = |5| - |3| = 5 - 3 = 2
In this case, |k - n| = |k| - |n| holds true.
2. If k and n are both negative integers:
Let's say k = -5 and n = -3.
|k - n| = |-5 - (-3)| = |-2| = 2
|k| - |n| = |-5| - |-3| = 5 - 3 = 2
Again, |k - n| = |k| - |n| holds true.
3. If k and n have opposite signs:
Let's say k = 5 and n = -3.
|k - n| = |5 - (-3)| = |8| = 8
|k| - |n| = |5| - |-3| = 5 - 3 = 2
In this case, |k - n| ≠ |k| - |n|, so the equation does not hold true.
Hence, we have found a counterexample where |k - n| ≠ |k| - |n|. Therefore, the statement |k - n| = |k| - |n| does not hold true for all integers k and n.