Solving Linear Inequalities

You already know that equations are mathematical statements that describe two expressions with equal values. When the values of the two expressions are not equal, their relationship can be described in an inequality.

Verbal phrases like greater than or less than describe inequalities. For example, 6 is greater than 2. This is the same as saying 2 is less than 6.

Addition and Subtraction Properties for Inequalities: For any inequality, if the same quantity is added or subtracted to each side, the resulting inequality is true.

Symbols: For all 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.

Multiplication Property for Inequalities: If you multiply each side of an inequality by a positive number, the inequality remains true. If you multiply each side of an inequality by a negative number, the inequality symbol must be reversed for the inequality to remain true.

Symbols: For all numbers a, b, and c,

if c is positive and a > b, then ac > bc, and if c is positive and a < b, then ac < bc.
if c is negative and a > b, then ac < bc, and if c is negative and a < b, then ac > bc.

Division Property for Inequalities: If you divide each side of an inequality by a positive number, the inequality remains true. If you divide each side of an inequality by a negative number, the inequality symbol must be reversed for the inequality to remain true.

Solving Linear Inequalities

Practice: Answer the following exercises by graphing inequalities on a number line.

Solving Linear Inequalities 1

Solving Linear Inequalities 2

Solving Linear Inequalities 3