Absolute Value..

Question

Which of the following statements is true for all integers?

1. ⏐m+n⏐≥⏐m⏐+⏐n⏐
2. ⏐m+n⏐≤⏐m⏐+⏐n⏐
3. ⏐m+n⏐<⏐m⏐+⏐n⏐
4. ⏐m+n⏐=⏐m⏐+⏐n⏐

Can you please explain how to approach such a problem?

Answer 1

So, let's think about our options. If m and n are both the same sign and we add them, we add the absolute values and keep the sign.

5 + 2 = 7 -----> |7| = 7

-5 + (-2) = -7 ---> |-7| = 7

Now, let's say they have different signs. We subtract the absolute values and keep the sign of the greater absolute value.

-5 + 2 = -3 ---> |-3| = 3

-2 + 5 = 3 ----> |3| = 3

Now, let's look at the right side.

|5| + |2| = 7 -----> 7 = 7

|-5| + |-2| = 7 ----> 7

|5| + |-2| = 7

|-5| + |2| = 7

So, the left side equals 3 or 7, but the right side always equals 7, sooo...

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