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The figure shows that (1, 5) ∪ (3, 7] = (1, 7]. Similarly, the union of three or more sets is the collection of objects that are contained in at least one of the sets. 1 5 3 7 30 chapter 1 The Real Numbers The next example goes in the other direction, starting with a set and then writing it as a union of intervals. example 3 Write the set of nonzero real numbers as the union of two intervals. solution The set of nonzero real numbers is the union of the set of negative numbers and the set of positive numbers.

Show that 67 + 2 is an irrational number. √ 2. Show that 5 − 2 is an irrational number. √ 3. Show that 3 2 is an irrational number. √ 3 2 5 is an irrational number. √ 5. Show that 4 + 9 2 is an irrational number. 4. Show that 6. Explain why the sum of a rational number and an irrational number is an irrational number. 7. Explain why the product of a nonzero rational number and an irrational number is an irrational number. 8. Suppose t is an irrational number. Explain why 1 is also an irrational number.

X + 3| = x + 3 solution Note that |x + 3| = x + 3 if and only if x + 3 ≥ 0, which is equivalent to the inequality 5 20 Thus [2, 7) ∪ [5, 20) = [2, 20). 21. [−2, 8] ∪ (−1, 4) solution The first interval is the set {x : −2 ≤ x ≤ 8}, which includes both endpoints. The second interval is {x : −1 < x < 4}, which does not include either endpoint. 3 Inequalities, Intervals, and Absolute Value 37 Thus [−2, 8] ∪ (−1, 4) = [−2, 8]. 23. (3, ∞) ∪ [2, 8] solution The first interval is {x : 3 < x}, which does not include the left endpoint and which has no right endpoint.

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