An inequality is a mathematical statement comparing two expressions using <, >, ≤, ≥, or ≠ instead of =. Solving linear inequalities follows the same rules as solving equations, with one critical exception: when you multiply or divide both sides by a negative number, you must reverse the inequality sign. For example, solving -2x > 6 requires dividing by -2, flipping > to <, giving x < -3.
Compound inequalities combine two inequalities joined by "and" or "or." An "and" inequality like -3 < 2x + 1 ≤ 7 means both parts must be true simultaneously. Solve by performing the same operations on all three parts: subtract 1 to get -4 < 2x ≤ 6, then divide by 2 to get -2 < x ≤ 3. The solution is the overlap of both conditions. An "or" inequality like x < -1 or x > 4 means the solution includes values satisfying either condition — the union of two separate intervals.
Graphing inequalities on a number line uses open circles for strict inequalities (< or >) and closed circles for inclusive ones (≤ or ≥). The line is shaded in the direction of all valid values. For example, x > 2 is graphed with an open circle at 2 and shading to the right. Compound inequalities form a segment on the number line: -2 < x ≤ 3 is an open circle at -2, a closed circle at 3, and shading between them.
Quadratic inequalities such as x² - 4 > 0 are solved by first finding the critical points (where the expression equals zero) and then testing intervals. Factoring x² - 4 = (x - 2)(x + 2) = 0 gives x = 2 and x = -2. These divide the number line into three intervals. Test a value from each: x = -3 gives 5 > 0 ✓, x = 0 gives -4 > 0 ✗, x = 3 gives 5 > 0 ✓. The solution is x < -2 or x > 2.