One variable inequalities are solved like equations. The key difference between equations and inequalities is that inequalities have a range or interval of solutions rather than an exact solution like an equation has. It is possible to have no solutions or infinite solution. Various examples of one variable inequalities are included.

Using Distributive Property

Solution-All Real Numbers

No Solution

Multistep

Multistep-Negative Coefficient

Solve with Fractions

Compound Inequalities

Compound inequalities contain statements like "or" or "and." In an "or" inequality, either one of the inequality statements must be true, not both. In an "and" inequality, both statements must be true. It is possible to have no solutions or to have all real numbers as the answer.

"Or" Compound Ex 1

"Or" Compound Ex 2

"And" Compound Ex 1

"And" Compound Ex 2

Absolute Value Inequalities

Absolute value inequalities have the variable inside of an absolute value. If the inequality is less than, an "and"
compound inequality is used. If the inequality is greater than, an "or" compound inequality is solved.

Absolute Value Less Than Ex 1

Absolute Value Greater Than Ex 1

Absolute Value Less Than Ex 2

Absolute Value Greater Than Ex 2

Polynomial Inequalities

Polynomial inequalities have a solution set that contains all values of the variable that make the inequality true. There could be one interval, multiple intervals or no intervals that make the polynomial true. These can be solved graphically or algebraically.

Solved Graphically

Solved Algebraically
Checked with TI-84

Solved Algebraically Ex 2 Checked with TI-Nspire

Rational Inequalities

Rational inequalities have a solution set that contains all values of the variable that make the inequality true. There could be one interval, multiple intervals or no intervals that make the polynomial true. These can be solved graphically or algebraically.

Solved Graphically

Solved Algebraically
Checked with desmos.com

Solved Algebraically Ex 2
Checked with desmos.com

Systems of Inequalities

Systems of inequalities are best solved by graphing. We are looking for the region of values that make both inequalities true.