A linear inequality is a mathematical expression that involves at least one linear function and one of the symbols of inequality: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), or ≠ (not equal to) . It looks exactly like a linear equation, but with the inequality sign replacing the equality sign. Linear inequalities can involve numerical or algebraic expressions or a combination of both.

To solve a linear inequality, the same procedure used for solving a linear equation is followed, with the exception that when multiplying or dividing both sides of the inequality by a negative number, the direction of the inequality switches. The solution set of a linear inequality can be represented using set builder notation, which is read as "the set of all x such that x satisfies the inequality".

Linear inequalities can be graphed on a number line to illustrate the solution set. For example, to graph the solution set of x + 3y < 9, one would first draw the line with equation x + 3y = 9 as a dotted line to indicate that the line is not included in the solution set since the inequality is strict. Then, a convenient point not on the line, such as (0,0), is chosen. Since 0 + 3(0) = 0 < 9, this point is in the solution set, so the half-plane containing this point (the half-plane "below" the line) is the solution set of this linear inequality.