1.26 Compound Inequality Definitions
Compound inequalities define ranges of values by combining two inequalities, often using 'and' or 'or', to express conditions in algebraic expressions.
Compound Inequality Definitions establishes the vocabulary for statements that combine two or more individual inequalities into a single condition, distinguishing the two ways such statements can be joined logically, describing the shorthand notation used when a variable is bounded on both sides, and defining the range of values that such a combined condition ultimately describes.
Compound Inequality
A compound inequality is a statement formed by combining two or more individual inequalities into a single condition that a variable must satisfy, typically joined by the logical connector "and" or "or." Compound inequalities allow a single statement to describe more complex sets of values than a single inequality alone, such as values restricted to fall between two boundaries, or values satisfying either of two separate conditions.
Conjunction
A conjunction is a compound inequality joined by "and," which is satisfied only by values that make every one of its individual inequalities true simultaneously. The compound inequality x > 2 and x < 7 is a conjunction, satisfied only by values of x that are simultaneously greater than 2 and less than 7, producing a solution set consisting of the numbers strictly between 2 and 7.
Disjunction
A disjunction is a compound inequality joined by "or," which is satisfied by any value that makes at least one of its individual inequalities true, without needing to satisfy both. The compound inequality x < 0 or x > 10 is a disjunction, satisfied by any value of x that is either less than 0 or greater than 10, producing a solution set consisting of two separate, unconnected ranges.
Chained Inequality
A chained inequality is a shorthand notation for a conjunction in which a single variable expression is bounded between two values, written as a single continuous statement rather than as two separate inequalities joined by "and." The chained inequality 2 < x < 7 is equivalent to the conjunction x > 2 and x < 7, expressing that x lies strictly between 2 and 7 in a single compact form.
Bounded Interval
A bounded interval is the range of values described by a chained inequality or an equivalent conjunction, containing every real number between two finite boundary values, with the endpoints included or excluded depending on whether the inequality symbols used are inclusive or strict. The chained inequality 2 < x < 7 describes the bounded interval consisting of all real numbers strictly between 2 and 7, excluding both endpoints.
Together, these definitions describe how simple inequalities are combined into richer conditions: conjunctions requiring every part to hold, disjunctions requiring at least one part to hold, chained inequalities offering a compact notation for a common type of conjunction, and bounded intervals representing the resulting finite range of values such combined conditions typically produce.