SECTION A.9 Interval Notation; Solving Inequalities A77 In each of these definitions, a is called the left endpoint and b the right endpoint of the interval. The symbol ∞ (read as “infinity”) is not a real number, but notation used to indicate unboundedness in the positive direction. The symbol −∞ (read as “negative infinity”) also is not a real number, but notation used to indicate unboundedness in the negative direction. The symbols ∞ and −∞ are used to define five other kinds of intervals: Note that ∞ and −∞ are never included as endpoints, since neither is a real number. Table 3 summarizes interval notation, corresponding inequality notation, and their graphs. Table 3 a b a a b a b b a a a a The open interval (a, b) The closed interval [a, b] The half-open interval [a, b) The half-open interval (a, b] The interval [a, q) The interval (a, q) The interval (- q, a] The interval (- q, a) The interval (- q, q) a 6 x 6 b a # x # b a # x 6 b a 6 x # b x $ a x 7 a x # a x 6 a All real numbers Inequality Graph Interval DEFINITION Intervals • An open interval , denoted by a b , ( ) , consists of all real numbers x for which a x b. < < • A closed interval , denoted by a b , [ ] , consists of all real numbers x for which a x b. ≤ ≤ • The half-open, or half-closed , intervals are a b , ( ] , consisting of all real numbers x for which a x b, < ≤ and a b , [ ) , consisting of all real numbers x for which a x b. ≤ < a, [ )∞ consists of all real numbers x for which x a ≥ . a, ( )∞ consists of all real numbers x for which x a > . a , ( ] −∞ consists of all real numbers x for which x a ≤ . a , ( ) −∞ consists of all real numbers x for which x a < . , ( ) −∞∞ consists of all real numbers. Writing Inequalities Using Interval Notation Write each inequality using interval notation. (a) x 1 3 ≤ ≤ (b) x 4 0 − < < (c) x 5 > (d) x 1 ≤ Solution EXAMPLE 1 (a) x 1 3 ≤ ≤ describes all real numbers x between 1 and 3, inclusive. In interval notation, we write 1, 3 [ ] . (b) In interval notation, x 4 0 − < < is written 4, 0 . ( ) − (c) In interval notation, x 5 > is written 5, . ( )∞ (d) In interval notation, x 1 ≤ is written , 1 . ( ] −∞ In Words An interval is a nonempty set of real numbers. In Words The notation a b , [ ] means all real numbers between a and b , inclusive. The notation a b , ( ) means all real numbers between a and b exclusive, that is, not including a and b . 1 Use Interval Notation Let a and b represent two real numbers with a b. <
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