Absolute Value Equations
Date: 10/16/98 at 22:24:55 From: Calvin DeRoo Subject: Absolute value in algebra The solution set is -3 < x < 11, and I'm supposed to make an equation out of it involving absolute value. I just don't know how to set this problem up right. Please help. I made a number line but I still don't know. Thank you.
Date: 10/16/98 at 22:52:54 From: Doctor Pat Subject: Re: Absolute value in algebra Calvin, The number line is a good first step. For absolute value problems it is easiest to think of center and distance. I'll explain. Look at the set of numbers on the number line you have described by -3 < x < 11. In English, we want to express all the numbers "between -3 and ll". To do that with absolute value, first find the number in the center. The midpoint of the segment of the number line from -3 to 11 is at 4. Now look at four, and the numbers in your set. Some of them are very close to 4, some are a little farther away, but how far are the most distant points? Well, 11 is seven units away, and -3 is also seven units away. So another way to describe these is by saying they are the numbers that are LESS THAN 7 units away from 4. Absolute value and distance are related because the distance of a number, x, on the number line away from the number 4 is given by |x-4|. The absolute value is needed because 3 and 5 are both 1 unit away, and without the absolute value we would get a distance of -1, which doesn't make sense in geometric distance (although we could use it as a VECTOR to tell us direction). From all this I hope you can see that to express the numbers centered at four and less than 7 units away, we could write |x-4| < 7 Can you see that: |x-6| < 2 would be the numbers between 4 and 8 |x-15| < 5 would be between 10 and 20 |x+5| < 3 would be between -8 and -2 because |x-(-5)| < 3 To show points farther than a given distance we would change the sign. The set x > 10 or x < 6 would be |x-8| > 2 since they are all MORE than 2 away from the center, 8. Hope that helps. Good luck. - Doctor Pat, The Math Forum http://mathforum.org/dr.math/
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