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A.9. Interval Notation; Solving Inequalities
Note. In this appendix, we introduce and use interval notation, give properties of inequalities, solve inequalities, and solve inequalities involving absolute value.
Definition. An inequality in one variable is a statement involving two expressions, at least one containing the variable, separated by one of the inequality symbols: < (less than), ≤ (less than or equal to), > (greater than), or ≥ (greater than or equal to). To solve an inequality is to find all values of the variable for which the statement is true. These values form the solution of the inequality.
Note. To solve inequalities, we need to introduce “interval notation.”
Definition. Let a and b be real numbers with a < b. Then
A closed interval, denoted by [a, b], consists of all real numbers x for which a ≤ x ≤ b. An open 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.
In each of these definitions, a is the left endpoint and b is the right endpoint of the interval.
Note. We will use the symbols ∞ (“infinity”) and −∞ (“negative infinity”) to indicate unboundedness in intervals. These are not numbers! If you take calculus, you will use these symbols extensively and, even then, they are not numbers! We have five kinds of unbounded intervals:
[a, ∞) consists of all real numbers x for which x ≥ a (i.e., a ≤ x < ∞). (a, ∞) consists of all real numbers x for which x > a (i.e., a < x < ∞). (−∞, a] consists of all real numbers x for which x ≤ a (i.e., −∞ < x ≤ a). (−∞, a) consists of all real numbers x for which x < a (i.e., −∞ < x < a). (−∞, ∞) consists of all real numbers x (i.e., −∞ < x < ∞).
Notice that neither ∞ nor −∞ are endpoints. The first four intervals have only one endpoint and the last interval has no endpoints.
Note. We can also draw pictures of intervals on the number line. From page A Table 2 we have:
Note. In dealing with inequalities involving absolute value, we use the following.
Theorem A.9.A. If a is any positive number, then:
|u| < a is equivalent to −a < u < a. |u| ≤ a is equivalent to −a ≤ u ≤ a. |u| > a is equivalent to u < −a or u > a. |u| ≥ a is equivalent to u ≤ −a or u ≥ a.
Examples. Page A80 numbers 98 and 100.
Example. Page A81 number 123.
Last Updated: 8/16/
