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# Prove some properties of the absolute value

Prove the following:

1. if and only if .
2. .
3. .
4. .
5. .
6. .
7. If then .
8. .
9. .
10. .

1. Proof. If then . If , then by definition of 2. Proof. If , then and .
If , then and (since 3. Proof. If , then , so and .
If , then , so and .
If , then so 4. Proof. First, . So, if then .
If then 5. Proof. Since is defined to be the unique non-negative square root of , from part (d) we have , and is non-negative by definition; hence, 6. Proof. If , then , so Further, , so .
If and , then so . Then, and , so If and , then so . Then and , so Finally, if and , then so . Then and , so 7. Proof. Using the definition of we have by part (f). But then since , we have . So, 8. Proof. Using the triangle inequality and the fact that , 9. Proof. Here we use the trick of adding and subtracting , and then the triangle inequality, 10. Proof. Using a similar trick to part (i), we have Then, combining this inequality with the one in part (i) and applying the definition of the absolute value, 1. Tiago says:

Actually I would make a small alteration: in part (f), you should have that if and < 0, then (just to be more precise)

2. Anonymous says:

hmm good….

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