Section 1.7 - Linear Inequalities

Section Objectives

1. Write inequalities corresponding to problem situations.
2. Write an interval using inequalities and graph it.
3. Solve linear inequalities.
4. Solve application problems involving linear inequalities.

A inequality is an algebraic expression involving less than ($\lt$), less than or equal to ($\le$), greater than ($\gt$), or greater than or equal to ($\ge$).

Using Inequalities to Describe Intervals

Inequalities are often used to describe intervals. Here are some examples.

Conditions	Set Notation	Interval Notation
$x$ is greater than $k$	$\{x \, | \, x > k\}$	$(k,\infty)$
$x$ is less than or equal to $k$	$\{x \, | \, x \le k \}$	$(-\infty,k]$
$x$ is less than $b$ and greater than $a$	$\{ x \, | \, a < x < b \}$	$(a,b)$
$x$ is less than $b$ and greater than or equal to $a$	$\{ x \, | \, a \le x < b \}$	$[a,b)$
$x$ is less than $a$ or $x$ is greater than $b$	$\{ x \, | \, x < a \mbox{ or } x > b \}$	$(- \infty, a) \cup (b, \infty)$

 

The last three examples are called compound inequalities. Compound inequalities are associated with the words "and" or "or".

To graph an inequality, we shade the interval(s) along a number line. Use parentheses ( ) to exclude endpoints, and use brackets [ ] include endpoints.

Solving inequalities

To solve an inequality means to find ALL replacements for the variable that make the inequality true. Your answers will typically (but not always) be intervals.

To solve a linear inequality, we apply the same process we used for solving linear equations, with one exception:

When solving a linear inequality, whenever you multiply (or divide) across by a negative number, you must reverse the inequality symbol(s).