Inequalities

Inequalities use four primary symbols to compare values: less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). These symbols show the relationship between two values that are not equal. Unlike equations that show equality, inequalities express a range of possible solutions rather than a single value.

Closed intervals use square brackets [a, b] and include both endpoints, while open intervals use parentheses (a, b) and exclude both endpoints. Half-open intervals like (a, b] or [a, b) include only one endpoint. When a set extends indefinitely, infinity symbols are used with parentheses, such as (a, ∞) or (−∞, b], since infinity is never included as an actual value.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality's direction. For example, multiplying both sides of 3 < 7 by −2 gives −6 > −14. This reversal is critical when solving inequalities to ensure the solution set remains accurate and the relationship between values is preserved.

The addition property states that adding or subtracting the same value from both sides of an inequality keeps the inequality's direction unchanged. If a < b, then a + c < b + c. This property allows you to isolate variables by removing constants from one side while maintaining the relationship between the two sides of the inequality.

Taking reciprocals reverses an inequality's direction only when both values are positive. If a < b and both are positive, then 1/a > 1/b. This reciprocal property does not apply when values are negative, making it essential to verify that both sides are positive before applying this operation during problem solving.

Inequalities model real-world constraints and restrictions. For example, a ride's height requirement stating 'over 100 cm tall' uses an inequality to define who can participate. These expressions help establish boundaries and ranges in practical scenarios, from safety regulations to resource allocation, making them essential for mathematical modeling problem solving.

Simplifying an inequality involves two main steps: first, remove the constant from one side by adding or subtracting, then divide both sides by the coefficient to isolate the variable. Remember that multiplying or dividing by a positive number preserves the inequality direction, while using a negative number reverses it. This process yields a solution set representing all values satisfying the inequality.