
Linear
Inequalities 


Linear
inequalities 
Inequality signs

Properties
of inequalities

Solving
linear inequalities 





Inequality signs

Inequality
is a statement about the relationship between two numbers or
quantities that holds when they are comparable and related by
ordering but are not equal. 
These
relationships are written as: 
a
< b
meaning
a
is
less than b, 
a ≤
b meaning
a
is
less than or equal to b, 
a
> b
meaning
a
is
greater than b, 
a ≥
b meaning
a
is
greater than or equal to b. 

Properties
of inequalities

The
trichotomy property of the real
line states that for any real numbers, a
and
b,
exactly one of 
a
< b,
a = b,
a
> b,
is true. 

The
transitive property of inequalities
states that for any real numbers, a,
b and
c, 
if a
< b
and b
< c,
then
a
< c, 
if a
> b and
b
> c,
then
a
> c. 

The
addition and subtraction property
of inequalities states that for any real numbers, a,
b and
c, 
if a
< b,
then
a
+ c
< b
+ c,
and a

c
< b

c, 
if a
> b,
then
a
+ c
> b
+ c, and
a

c
> b

c. 

The
multiplication and division
property of inequalities states that for any real numbers, a,
b and
c, 
if c
is
positive and
a
< b,
then
a
· c
< b
· c,
and a/c
< b/c, 
if c
is
negative and
a
< b,
then
a
· c
> b
· c,
and a/c
> b/c. 

The
sense of an inequality is not changed if both sides are
increased or decreased by the same number, or if both sides are
multiplied or divided by a positive number. 
The sense of an
inequality is reversed if both sides are multiplied or divided
by a negative number. 

The
additive inverse property of inequalities states that for any real
numbers a
and
b, 
if
a
< b
then
a
>
b, 
if
a > b
then
a
< b. 

The
multiplicative inverse
property of inequalities states that for any real numbers a
and
b
that are both positive or both negative, 

if
a
< b
then
1/a
>
1/b, 
if
a > b
then
1/a
<
1/b. 

Note that, the above properties also hold if strict inequality signs, < and >, are replaced with their corresponding non strict
inequality signs, ≤
and ≥. 

Solving
linear inequalities

A linear inequality has the standard form
ax +
b
> 0, where a,
b
Î
R 
and where other inequality
signs like <, > and < can appear. 
As with equations, the inequality is solved when positive x is isolated on the left. 
The solutions to an inequality are all values of
x
that make the inequality true. 
We may add the same number to both sides of an inequality. 
By multiplying 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. 

Example: 

x 
8
< 2x

4 
 x
<
4

·
(1) 
x
>
4 


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











Beginning
Algebra Contents C 



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