Section 41
Solving Inequalities in One Variable
Just as
equations
are mathematical sentences which state that two quantities are
equal
,
inequalities
are sentences which state that two quantites are
not equal
(or possibly not equal). There are five inequality symbols:
x
≠ 3

x
is not equal to 3

x
< 3

x
is less than 3

x
> 3

x
is greater than 3

x
3

x
is less than or equal to 3

x
3

x
is greater than or equal to 3

Often, for the last four types of inequalities, we need to
solve the inequality
so that the variable is alone on one side. This is done using analogues of the
properties of equality
: adding or subtracting the same quantity to both sides, or multiplying or dividing both sides by the same quantity. The only
important difference
is that:

Whenever you multiply or divide both sides of an inequality by a
negative number
, you need to
reverse
the direction of the inequality.
Example:
Solve for
x
.
–3
x
+ 2
14
First, subtract 2 from both sides.
–3
x
12
Now divide both sides by –3.
Remember to reverse the inequality.
x
–4
GRAPHING INEQUALITIES
The solutions of inequalities can be graphed on the number line as rays. If the inequality is "strict" (< or >), we use an open dot to indicate that the endpoint of the ray is not part of the solution. For the other types of inequalities (
and
), we use a closed dot.
Example: