

ALGEBRA


::
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.

Method of solving
rational inequalities

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.

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).


Example: 
Solve the rational
inequality



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. Therefore, we have to solve the 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, 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.


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].



::
Absolute value
inequalities

Graphical
interpretation of the
definition of the
absolute value of a function y =
f
(x) will help us solve
linear inequality with absolute value. The
definition for the absolute value of a function is given by


Solving
linear inequalities with absolute value

Thus,
for
values of x
for which f
(x)
is nonnegative, the graph of  f
(x) is the
same as that of f
(x).

For values of
x
for which f (x)
is
negative, the graph of  f
(x)
 is a reflection of
the graph of f (x)
on the xaxis.
That
is, the
graph of y = 
f
(x) is obtained by reflecting the graph
of y = f
(x)
across the xaxis.



::
Systems
of linear inequalities

Solving and graphing systems of
linear inequalities

The solution of the system of
simultaneous inequalities is the intersection of sets
of 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 ). 





::
Systems of
linear inequalities in two variables

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.



Plug the
coordinates of the origin 






::
Quadratic
inequalities

To solve a quadratic inequality we
examine signs of the
equivalent quadratic function.

The
xintercepts
or roots are the points where a quadratic function changes the sign. The
xintercepts determine
the three intervals on the xaxis in which the function is above or
under the xaxis, that is, where the function is positive or negative.

Graphic solution of
quadratic inequalities


Example: 
Solve the inequality

x^{2}
+ 2x
+
3 ≤
0.


Solution:
Solve the quadratic
equation ax^{2}
+ bx
+
c
= 0
to get the boundary points.

The zeroes
or roots of equivalent function (see the graph
below) are the endpoints of the intervals and are included in the solution.

The turning point V(x_{0},
y_{0}),
V(1,
4) 

The roots of, 
x^{2}
+ 2x
+
3 =
0 



Solution: 



















Contents A






Copyright
© 2004  2020, Nabla Ltd. All rights reserved.


