abs_value

# abs_value - A reminder about absolute values Math 8 2009...

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Unformatted text preview: A reminder about absolute values Math 8 2009, Chris Lyons • For x ∈ R , we define the absolute value of x to be | x | = x if x ≥- x if x < . • From this definition, one can show that if C ≥ 0 then | x | ≤ C ⇐⇒ - C ≤ x ≤ C. How do you show this? Here’s one direction, namely | x | ≤ C = ⇒ - C ≤ x ≤ C : Assume that | x | ≤ C . There are two cases: 1. Assume x ≥ 0; then automatically we have x ≥ - C , and if we know that | x | ≤ C , then by definition (because x ≥ 0) this means x ≤ C . Thus we get both x ≥ - C and x ≤ C , meaning- C ≤ x ≤ C . 2. On the other hand, if x < 0, then automatically we get x ≤ C , and from | x | ≤ C we get | x | =- x ≤ C , or x ≥ - C . Thus, no matter what x is, if | x | ≤ C , then we have- C ≤ x ≤ C . I’ll leave it to you to show that- C ≤ x ≤ C = ⇒ | x | ≤ C . (Again, there are two cases to consider: (1) x ≥ 0 and (2) x < 0.) • Using the previous point, if C ≥ 0 and x, y ∈ R , then we have...
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abs_value - A reminder about absolute values Math 8 2009...

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