
ALGEBRA
 solved problems 





Linear
inequalities

Solving linear
inequalities 



Solve given linear inequality.


Solution: 

x 
8
< 2x

4 
 x
<
4

·
(1) 
x
>
4 


The
solution of the inequality is every real number greater
than 4. 



87. 
Solve inequality
3(x 
2) >
2(1
x).


Solution:
3x 
6
>
2
+
2x 
x > 4 


interval
notation
(4, oo) 


The
open interval (4, oo)
contains all real numbers between given endpoints, where round
parentheses indicate exclusion of endpoints. 

88. 
Solve given linear inequality.


Solution:


4x
+
9 
3x
<
6  5
+
5x 
12x
<
8 
x
>
2/3 



interval
notation 



The
halfclosed (or halfopen) interval
contains all real numbers between given endpoints, where the
square bracket indicates inclusion of the endpoint 2/3 and round
parenthesis indicates exclusion of infinity. 

89. 
Solve inequality
(x
 3)
· (x
+ 2)
> 0


Solution:
The factor
x
 3
has the zero at x
= 3, is negative for x
< 3
and is positive for x
> 3,
and 
the factor
x
+ 2
has the zero at x
= 2,
is negative for x
<  2
and is positive for x
> 2, 
as
is shown in the table 
x 

oo 
increases 
2 
increases 
3 
increases 
+
oo 
x
+ 2 

 
0 
+ 
+ 
+ 

x
 3 

 
 
 
0 
+ 

(x
 3)(x
+ 2) 

+ 
0 
 
0 
+ 



Thus,
the given inequality is satisfied for 
oo
< x
<  2
or 3
< x
< + oo 
in the interval notation (
oo
, 2]
U [3,
+ oo
) 


Solving
compound (double) inequalities


90. 
Solve inequality
4
< 2(x
 3)
< 5


Solution:
We want the x alone as middle term and only constants in the two outer terms. Remember,
while simplifying given compound inequality, the operations that we apply to a middle term we
should also do to the both left and right side of the inequality. 





Solve given linear inequality.





Systems
of linear inequalities

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


92. 
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.


93. 
Solve and graph the
solution of the given system of linear inequalities in two
variables.


Solution:


Plug the
coordinates of the origin 














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