

Rational
Inequalities 
The
graph of the translated equilateral (or rectangular) hyperbola






The graph of the
translated equilateral (or rectangular) hyperbola

The graph of the given rational function is translated equilateral (or rectangular)
hyperbola.

A
rational function of the
form


can be rewritten into




where 

the vertical asymptote,


the horizontal
asymptote 



and the
parameter




Therefore, values of the vertical and
the horizontal asymptote correspond to the coordinates of the horizontal and the vertical translation
of the source
equilateral hyperbola y
= k/x, respectively.

Thus, given rational function 

where, a
= 1, b
=
2
and c
=
1,
d
=
1 


has
the vertical asymptote 




the
horizontal asymptote 




and
the parameter 



Therefore,
its source function is the equilateral or rectangular
hyperbola 



The graph of given
rational function is shown below.



Example:
Find the solutions
of the inequality




Solution:
The solution set of the given rational inequality includes all numbers
x
which make the inequality less then or equal to 0, or which make the sign of the rational expression
to be negative or 0.

A rational expression is negative if the numerator and the denominator
have different signs, and the rational expression equals 0 when
its numerator is equal to 0 that is,


therefore, we have to solve two simultaneous
inequalities.

We graph the numerator and the
denominator in the same coordinate system to find all points of
the xaxis
that satisfy given inequality.

The zero points of the numerator and
the denominator divide the xaxis
into four intervals at which given rational expression changes
sign.

x^{2}
+ 2x 
3 = 0, a = 1, b = 2 and
c = 3






The solutions of the two pairs of the simultaneous
inequalities are intersections of sets of their partial solutions,


as is shown below.


Therefore, the solution set is (
oo,
 3]
U
(
2,
1].









Intermediate
algebra contents 



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