Inequalities

Inequalities express mathematical relationships where two values are not equal and are compared using symbols such as <, >, ≤, or ≥. These expressions define a range of possible solutions rather than a single value. Interval notation provides a concise way to express these solution sets, especially when the variable spans a continuous range. An open interval, written as (a, b), excludes the endpoints, while a closed interval [a, b] includes them. There are also half-open intervals, such as (a, b] and [a, b), which include only one endpoint. The notations (a, ∞) and (−∞, b] are used when a set extends indefinitely in one direction, with infinity symbols never being included as actual values.

Solving inequalities requires applying operations while adhering to specific rules that maintain or reverse the direction of the inequality. When a constant is added or subtracted from both sides, the inequality’s direction remains unchanged. This is formalized in the addition property: if
a < b, then a + c < b + c. Similarly, multiplying or dividing both sides by a positive number retains the inequality's direction, whereas using a negative number reverses it. For example, multiplying both sides of 3 < 7 by 2 results in 6 < 14, but multiplying both sides of 3 < 7 by −2 gives -6 > -14.

The reciprocal property of inequalities also plays a critical role. If a > 0, then 1/a > 0, and taking reciprocals of both sides of an inequality involving positive values reverses the inequality’s direction. For instance, if a < b and both are positive, then 1/a > 1/b.