
ALGEBRA
 solved problems 






Rational
Inequalities



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
).

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, (see the above example of the rational inequality).




Solve the rational
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].


Absolute value
inequalities 
Solving
linear inequalities with absolute value 
Graphical
interpretation of the
definition of the
absolute value of a function y =
f(x) will help us solve
linear inequality with absolute value. 

96. 
Solve the absolute value
inequality  x

2
 < 3.


Solution:
Graphical
interpretation of the given inequality will help us find
values of x
for which left side of the inequality is less then or equal to
3. 
For
values of x
for which y
is nonnegative, the graph of  y
 is the
same as that of y =
x 
2.

For values of
x
for which y
is
negative, the graph of  y
 is a reflection of
the graph of y
across the x
axis. 
Since
the graph of y =
x 
2 has y
negative on
the interval (
oo, 2) it is this part of the graph
that has to be reflected on the x
axis. 

The
graph shows that
values of x from
the closed interval [1,
5] satisfy the given inequality. 
The
same result can be obtained algebraically by solving the
compound inequality. 


x
Î [1,
5] 



97. 
Solve the absolute value
inequality  x

2
 > 3.


Solution:
Again, graphical
interpretation of the given inequality will help us find
values of x
for which left side of the inequality is greater than 3. 
The
graph shows that
values of x from
the open intervals, ( oo, 1)
or (5,
oo)
satisfy the given inequality. 

The
same result can be obtained algebraically by solving the two
inequalities 











Solved
problems contents 


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