169 1.7 Inequalities 0 4 Figure 9 A graph of the solution set is shown in Figure 9, where the parenthesis is used to show that 4 itself does not belong to the solution set. As shown below, testing values from the solution set in the original inequality will produces true statements. Testing values outside the solution set produces false statements. CHECK -3x + 5 7 -7 Original inequality The solution set of the inequality, 5x x 646, Set-builder notation is an example of an interval. We use interval notation to write intervals. With this notation, we write the above interval as 1-∞, 42. Interval notation The number 4 is an endpoint of the interval. The symbol -∞ is not an endpoint because it does not represent an actual number. Rather, it is used to show that the interval includes all real numbers less than 4. The interval 1-∞, 42 is an example of an open interval because it does not contain any of its endpoints. An interval that includes all its endpoints is a closed interval. A square bracket to the left or right of an endpoint indicates that the number is part of the interval, and a parenthesis indicates that the number is not part of the interval. S Now Try Exercise 13. In the table that follows, we assume that a 6b. Summary of Types of Intervals Set-Builder Notation Verbal Description Graph Interval Notation 5x x 7a6 The set of real numbers greater than a a 1a, ∞2 5x x Ú a6 The set of real numbers greater than or equal to a a 3a, ∞2 5x x 6b6 The set of real numbers less than b b 1∞, b2 5x x … b6 The set of real numbers less than or equal to b b 1-∞, b4 5x a 6x 6b6 The set of real numbers between a and b a b 1a, b2 5x a … x 6b6 The set of real numbers greater than or equal to a and less than b a b 3a, b2 5x a 6x … b6 The set of real numbers greater than a and less than or equal to b a b 1a, b4 5x a … x … b6 The set of real numbers between a and b, inclusive b a 3a, b4 5x x 6aor x 7b6 The set of real numbers less than a or greater than b b a 1∞, a2 ´1b, ∞2 5x x is a real number6 The set of all real numbers 1-∞, ∞2 -3102 + 5 7 ? -7 Let x = 0. 5 7 -7 ✓ True -3152 + 5 7 ? -7 Let x = 5. -10 7 -7 False
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