Question

For any two integers x and y,

|x+y| = |x| + |y|

True or False? If false, prove it.

Answer 1

Answer: False

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Step-by-step explanation:

Let's pick a positive number for x and negative number for y.

I'll pick x = 2 and y = -3 as the counter-example.

|x+y| = |x| + |y|

|2 + (-3)| = |2| + |-3|

|2 - 3| = 2 + 3

|-1| = 5

1 = 5

We get a false statement at the end, which means the original equation is false when (x, y) = (2, -3)

I'll let you try other counter-examples.

If both x and y are positive together, or negative together, then the original equation would always be true.

If x and y are mixed (one positive the other negative), then the statement is always false.

Overall |x+y| = |x| + |y| is false for any two integers x and y.