“Anytime somebody is absolutely certain about something, they are almost always absolutely wrong.”
~ Jeff Lindsay
Absolute values are weird.
The popular way of handling them is, “Get rid of all the signs.”
And if you’re taking the absolute value of a real number, that works for the most part.
But if you’re working with variables, it might not.
Working with Numbers
Suppose I ask you:
- Pick your favorite number.
- Subtract 100 from it.
- Take off the sign (so if it’s negative, make it positive).
You now have a positive number. But at step 2, you probably didn’t. (The top 10 favorite numbers are all pretty small.)
Using the world’s most popular favorite number (7), we would do this:
- Pick your favorite number: 7
- Subtract 100 from it: 7-100 = -93
- Take off the sign: 93
Working with Letters
Now let’s look at an expression: |x-100|
The absolute value bars indicate to “take off the sign” – our popular definition.
Many people make |x-100| into x+100.
Compare this to our example above:
- Pick your favorite number. (We’ll call it x…)
- Subtract 100 from it. (…so that would be x-100…)
- Take off the sign. (…or do |x-100|.)
For our number 7, we got 93.
But if we claim that |x-100| is the same as x+100, then 93 is the same a 7+100.
Everything You Know is Wrong
Okay, no it’s not. (But that’s a great Weird Al song!)
So you have to “do the inside first,” as they say.
Which means you can’t just get rid of them that easily.
The Bright Side
People believe that absolute value bars turn all negatives into positives.
This is a pretty good way to think, at least in life.
So the next time you’re feeling yucky, grab a couple of sticks or pool noodles and hold them to either side of you.
Now smile – you’re positive!
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