Algebra 1 Equation Manipulation
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Fancy Onions
Equation Manipulation

 

 Back on the page "Peel the Onion" we listed 2 rules

 for peeling stuff away from the X.
 
 First peel away any term on the side with the X
 that doesn't have an X in it.
 
 If there's only one term on the side with the X,
 stuff on the other side of the biggest fraction line
 is the stuff that's farthest away from the X.
 
 Since that page, we have done a lot more shenanigans with X's.
 To keep our Onion Peeling rules up to date
 we need to add another rule ...
 
 If there is only one term on the side with the X
 and the X is somewhere inside the parenthesis,
 deal with the stuff on the outside of the parenthesis first.
 
 Now we have 3 rules.
 That's all the new rules, for now anyway.
 The new rule goes between the other two.
 Think of these rules like step by step instructions.
 
 Rule 1: First peel away any term on the side with the X
 that doesn't have an X in it.
 
 Rule 2: If there is only one term on the side with the X,
 and the X is somewhere inside the parenthesis,
 deal with the stuff on the outside of the parenthesis first.
 
 Rule 3: If there is only one term on the side with the X,
 stuff on the other side of the biggest fraction line
 is farthest away from the X.
 
 We have used parenthesis before
 but there is one special parenthesis that we need to talk about.
 It wears a disguise.
 The disguise looks like this:
 

 

 
 Enough already with all this silly theory.
 Let's do some problems !!!
 
 Example:
 
 Solve for X ...

 

 
 We start with rule 1.
 It says that any term that has no X in it
 on the same side of the equation as the X
 is farthest away. Peel it away first.
 
 That means we start with the "+ 2".
 To get rid of a "+ 2" we subtract 2.
 Anything we do to one side, 
 we need to do to the other side too.
 

 
 Now we have one term on the side with the X.
 The X is somewhere inside of the parenthesis
 so we deal with the stuff on the outside 
 of the parenthesis first.
 
 We can get rid of an exponent of 2
 by taking the square root.
 Remember that we need to do the same thing
 to both sides of the equation ...
 

 
 Now we have two terms on the side with the X again.
 So we go back to rule 1.
 We need to peel away a "- 4"
 so we add 4 to each side ...
 

 
 Example:

 

 
 First we peel away the term with no X in it ...
 

 
 Now we have one term with that sneaky parenthesis
 around everything on the side with the X.
 
 There is nothing outside of the sneaky parenthesis,
 so we deal with the sneaky parenthesis itself.
 The way to get rid of a square root,
 is to square it.
 Don't forget we need to do that 
 to both sides of the equation ...
 

 

 
 Now we have one term on the side with the X.
 There are no parenthesis.
 We go to rule 3.
 The thing on the far side of the fraction line
 from the X is a 4.
 A 4 in the denominator is like 1/4
 To peel it away, we multiply by 4/1 ...
 

 

 
 Now we are back to rule 1.
 Peel the "+ 3" away from the X.
 Subtract 3 from each side ...
 

 

 
 OK. Let's do one more ... 

 

 Example:

 

 Yuk! What a mess!
 Just take it one step at a time
 and everything will work out.
 
 First peel away the term with no X.
 

 
 OK, now we have 1 term with parenthesis.
 But what's the furthest outside?
 The 8 or the squared (the 2)?
 
 Here's the deal.
 The 2 is an exponent.
 An exponent is an instruction that says:
 "multiply the thing on my left times itself
 this many times."
 
 That means the problem  can be written like this:
 

 

 
 The 8 is the thing that is not part of the X.
 That means it goes away first.
 
 The 8 is multiplied times the other stuff
 so we get rid of it by dividing ...
 

 
 This looks a lot like the last example now.
 
 Since we have parenthesis, we need rule 2.
 We get rid of an exponent of 2, 
 by taking the square root of each side ...
 

 
 Now we have one term with no parenthesis.
 We do have a denominator, so it's time for rule 3.
 

 
 Now we have 2 terms again.
 That means it's time for rule 1.
 We have a "- 1" to peel away,
 so we do it with a "+ 1" on each side ...
 

 

   copyright 2005 Bruce Kirkpatrick

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