Math Help for Section 3.6, Page 146

Solving Linear Inequalities
An inequality in one variable is a linear inequality if it can be written in one of
the following forms.

    $ax+b\le0$,    $ax+b<0$,    $ax+b\ge0$,    $ax+b>0$

    Solving a linear inequality is much like solving a linear equation. You
isolate the
variable by using the properties
of inequalities. These properties are similar to
the properties of equality, but there are two important exceptions. When each
side of an inequality is multiplied or divided by a negative number,
the direction
of the inequality symbol must be reversed. Here is an
example.

    $\eqalign{-2<&5
&{\small\color{red}\quad\quad\text{Original inequality}} \cr
{\color{red}(-3)}(-2)>&{\color{red}(-3)}(5)
&{\small\color{red}\quad\quad\text{Multiply each side by
}-3\text{ and reverse the inequality.}}
\cr 6>&-15
&{\small\color{red}\quad\quad\text{Simplify.}} }$

    Two inequalities that have the same solution set are equivalent inequalities.
The list of operations on page 80 can be used to create equivalent
inequalities. These properties remain true when the symbols $<$ and
$>$ are replaced by $\le$ and $\ge$.
Moreover, a,
b, and c can represent
real numbers, variables, or expressions. Note
that you cannot multiply or divide each side of an inequality by zero.

Study Tip
The solution set of a linear inequality can be written in set notation. For instance, the
solution $x>1$ is written in set notation as $\{x|x>1\}$ and is
read “the set of all x
such that x
is greater than 1.”

Example 3: Tip
The solution set in set notation is $\{x|x<3\}$.