

Linear Inequalities

Solving inequalities 
Properties of
inequalities 
Examples
of solving single linear inequalities 
Solving
compound (double) inequalities 






Linear
inequalities

A linear inequality
is one that can be reduced to the standard form ax +
b
> 0
where a,
b
Î
R,
and where other inequality signs like <,
>
and <
can appear. 
Solving
inequalities 
The solutions to an inequality are all values of
x
that make the inequality true. Usually the answer is a range of
values of x
that we plot on a number line. 
We
use similar method to solve linear inequalities as
for solving linear equations: 

simplify both sides, 

bring all the terms with the variable on one side and the
constants
on the other side, 
 and then multiply/divide both sides by the
coefficient of the variable to get the solution while applying following properties: 

Properties of
inequalities 
1. Adding
or subtracting the same quantity from both sides of an
inequality will not change the direction of the inequality sign. 
2. Multiplying
or dividing both sides of an inequality by a positive number leaves the
inequality symbol unchanged. 
3.
Multiplying or dividing both sides of an inequality
by the same negative number, the sense of the inequality changes,
i.e., it reverses the direction of the inequality sign. 

Examples
of solving single linear inequalities 
Solve
each of the following inequalities, sketch the solution on the
real number line and express the solution in interval notation. 
Example: 

x 
8
< 2x

4 
 x
<
4

·
(1) 
x
>
4 


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



Example:
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. 

Example:


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. 

Example:
(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 or double inequalities 
Use the same procedure to solve a compound inequality as for solving single inequalities. 
Example:
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. 




Example: 


Solution: 










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