

Systems
of Linear Inequalities

Solving and graphing systems of
linear inequalities

Solving and graphing systems of
linear inequalities in two variables

Rational
Inequalities 
Method of solving
rational inequalities

The
graph of the translated equilateral (or rectangular) hyperbola






Systems
of Linear Inequalities

The solution of the system of
simultaneous inequalities is the intersection of sets
of the individual solutions.

Example:
Solve and graph the
solution of the given system of
simultaneous inequalities.





the solution of the system
is the open interval (3, +
oo
). 


Solving and graphing systems of
linear inequalities in two variables

The set of points whose coordinates (x,
y)
satisfy the inequality ax+
by + c > 0 is a
halfplane of a Cartesian plane.

Example:
Solve and graph the
solution of the given system of linear inequalities in two
variables.

Solution:


Plug the
coordinates of the origin 





Thus, non shaded area of the Cartesian
plane where lies the origin, bounded by given lines, is the solution of the system of inequalities.


Rational
Inequalities

A rational inequality can be written
in one of the following standard forms:

P/Q > 0 or P/Q <
0 (or P/Q > 0
or
P/Q < 0 ), where
Q is
not 0.

The sign of the rational expression P/Q, where P and Q are polynomials, depends on the signs of P and Q.

As the signs of the
polynomials change at the zeros, to solve a rational inequality we should find the zeros of

both P and Q first and then we can determine the intervals of the independent variable that satisfy given

rational inequality.


Method of solving
rational inequalities

The first step to solve a rational
inequality is to get a single rational inequality on the left side
of the inequality sign and have zero on the right side of the
inequality sign.

The next step is to factor the numerator and denominator and find the values of
x
that make these factors equal to 0 to find critical points (boundaries, endpoints of intervals).

Note that, by setting the numerator to
0 we get the zero points of the given rational expression, but by setting

the denominator
to 0 we get points at which the rational expression or function is undefined
(i.e., when

plugged into the expression give division by zero).

The rational functions are not defined at the zeros of the denominator. Therefore, they have breaks or vertical

asymptotes at these points,
and this is why these points cannot be included into the solution of the rational

inequality.


Example:
Solve the rational
inequality


and
draw the graph of the rational function. 

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

A rational expression is positive if both the numerator and the denominator are positive
or if both are negative, and the rational expression equals 0 when
its numerator is equal to 0 that is


therefore, we have to solve two simultaneous
inequalities:


The solutions represented on the
number line are shown below.


Thus, the solution set of the given
inequality written in the interval notation is (
oo,
1)
U
[2,
oo
).


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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