



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 
