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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? Heres 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 . Ill 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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