Is the sum of two integers always greater than the difference between them? Why or why not? Give examples to support your answer.
I think the answer is no.
You're right.
22 + 23 is greater than 1
To determine whether the sum of two integers is always greater than the difference between them, we can analyze it through mathematical reasoning.
Let "a" and "b" be two integers. The sum of these two integers is expressed as a + b, and the difference between them as a - b.
To prove that the sum is always greater than the difference, we would need to show that:
a + b > a - b
We can start by expanding the right side of the inequality:
a + b > a - b
a + b > a + (-b)
Now, we can cancel out the common term "a" from both sides:
b > -b
Since "b" and "-b" are additive inverses (when added, they result in zero), the above inequality represents that "b" is always greater than its additive inverse "-b". This statement is always true, except when b = 0.
Therefore, for any two integers a and b, the sum a + b will indeed be greater than the difference a - b, except when b = 0.
Example 1:
Let's take a = 3 and b = 5.
Sum: 3 + 5 = 8
Difference: 3 - 5 = -2
In this case, the sum (8) is greater than the difference (-2).
Example 2:
Let's consider a = 2 and b = -4.
Sum: 2 + (-4) = -2
Difference: 2 - (-4) = 6
Again, in this case, the sum (-2) is less than the difference (6).
These examples support the conclusion that the sum of two integers is not always greater than the difference between them.