Inequalities

Inequalities are mathematical expressions involving the symbols <, >, ≤, or ≥, which stand for "less than", "greater than", "less than or equal to", and "greater than or equal to", respectively.

Definition

An inequality is a relation between two expressions that may not be equal. The basic symbols used in inequalities are:

'<': less than
'>': greater than
'≤': less than or equal to
'≥': greater than or equal to
'≠': not equal to

For example, the inequality x > 3 means that x is greater than 3.

Solving Inequalities

Solving inequalities involves finding all the values of the variable that make the inequality true. The steps for solving inequalities are similar to those for solving equations, with one important difference: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.

Graphing Inequalities

Inequalities can be graphed on a number line or a coordinate plane. On a number line, an open circle is used to represent 'less than' or 'greater than', and a closed or filled-in circle is used to represent 'less than or equal to' or 'greater than or equal to'. On a coordinate plane, inequalities are represented by shaded regions.

Applications

Inequalities are used in a wide variety of fields, including mathematics, physics, engineering, economics, and computer science. They are used to model and solve problems where there is a range of possible solutions, rather than a single solution.