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Non-linear
Inequalities
The
procedures to solve a non linear inequality:
1) Move all
terms to the left side of inequality and leave 0 as the only number
on the right side.
2) Factor the
expression on left side completely; and interpret the inequility
as a "sign" problem.
3) Find all
possible intervals determined by its real zeros of the expression
plotted on a real number 
line.
4) Choose a
test number in each interval and evaluate the expression for each
number and mark the sign
on that interval. Thus we have a sign chart on the real number line.
5) Write down
the answer fron the sign chart that satisfies the inequality.
6) If there
is more than one interval to be included, then solution is the union
of all of them.
Ex
1: Solve
Transposing
all term to the left member we get 
Factoring the
left member we get 
This means the
product  has
to be negative. Let's study the sign of for
all each real
number.
First determine
the "critical values", that is, the values where the expression
is
equals to zero.
Secondly, mark
this number on the real line with a vertical line segment
____________________|______________|______________
0
1
2 3
4
Now we have
three interval as possible solution, so we'll check each one.
For interval
 we
select a checking point, for example x = 0. Replacing on we
get (0-1)(0-3)= +3, that is positive (so we reject
this interval)
For interval
(1,3) we select a checking point, example x = 2. Replacing
on we
get (2-1)(2-3)= -1, that is negative (so we accept
this interval as a solution).
For interval
 we
select as checking point any point in it, for example x = 4. Replacing
on we
get (4-1)(4-3)= +3, that is positive (so we reject
this interval)
Resuming we
have:
___________________1_____________3_______________
SIGN
+ + + + + + + + 0 - - - - -
- - - 0 + + + + + + + +
So the solution
is the interval (1 , 3)
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