5.6 Linear inequalities in one variable

Overview

• Inequalities
• Linear Inequalities
• The Algebra of Linear Inequalities
• Compound Inequalities

Inequalities

Relationships of inequality

Linear inequalities

Linear inequality

Inequality notation

Note that the expression $x>12$ has infinitely many solutions. Any number strictly greater than 12 will satisfy the statement. Some solutions are 13, 15, 90, $12.1,16.3\text{and}102.51$ .

Sample set a

The following are linear inequalities in one variable.

The following are not linear inequalities in one variable.

Practice set a

A linear equation, we know, may have exactly one solution, infinitely many solutions, or no solution. Speculate on the number of solutions of a linear inequality. ( Hint: Consider the inequalities $x<x-6$ and $x\ge 9$ .)

A linear inequality may have infinitely many solutions, or no solutions.

The algebra of linear inequalities

Inequalities can be solved by basically the same methods as linear equations. There is one important exception that we will discuss in item 3 of the algebra of linear inequalities.

The algebra of linear inequalities

1. $\begin{array}{ccc}a+c<b+c& \text{and}& a-c<b-c\end{array}$ .
   If any real number is added to or subtracted from both sides of an inequality, the sense of the inequality remains unchanged.
2. If $c$ is a positive real number, then if $a<b,$
   $\begin{array}{ccc}ac<bc& \text{and}& \frac{a}{c}<\frac{b}{c}.\end{array}$
   If both sides of an inequality are multiplied or divided by the same positive number the sense of the inequality remains unchanged.
3. If $c$ is a negative real number, then if $a<b,$
   $\begin{array}{ccc}ac>bc& \text{and}& \frac{a}{c}\end{array}>\frac{b}{c}.$
   If both sides of an inequality are multiplied or divided by the same negative number, the inequality sign must be reversed (change direction) in order for the resulting inequality to be equivalent to the original inequality. (See problem 4 in the next set of examples.)