EXAMPLE 3 Use inductive and deductive reasoning

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EXAMPLE 3 Use inductive and deductive reasoning ALGEBRA What conclusion can you make about the product of an even integer and any other integer? SOLUTION STEP 1 Look: for a pattern in several examples. Use inductive reasoning to make a conjecture. (–2) (2) = –4, (–1) (2) = –2, 2 (2) = 4, 3 (2) = 6, (–2) (–4) = 8, (–1) (–4) = 4, 2 (–4) = –8, 3 (–4) = –12 Conjecture: Even integer Any integer = Even integer

EXAMPLE 3 Use inductive and deductive reasoning STEP 2 Let: n and m each be any integer. Use deductive reasoning to show the conjecture is true. 2n is an even integer because any integer multiplied by 2 is even. 2nm represents the product of an even integer and any integer m. 2nm is the product of 2 and an integer nm. So, 2nm is an even integer. The product of an even integer and any integer is an even integer. ANSWER

EXAMPLE 4 Reasoning from a graph Tell whether the statement is the result of inductive reasoning or deductive reasoning. Explain your choice. a. The northern elephant seal requires more strokes to surface the deeper it dives. b. The northern elephant seal uses more strokes to surface from 250 meters than from 60 meters.

EXAMPLE 4 Reasoning from a graph SOLUTION Inductive reasoning, because it is based on a pattern in the data a. b. Deductive reasoning, because you are comparing values that are given on the graph

GUIDED PRACTICE for Examples 3 and 4 Use inductive reasoning to make a conjecture about the sum of a number and itself. Then use deductive reasoning to show the conjecture is true. 5. SOLUTION Conjecture: The sum of a number and itself is twice the number. Deductive reasoning: Let n be any integer. Use deductive reasoning to show the conjecture is true n + n = 2n 

GUIDED PRACTICE for Examples 3 and 4 Use inductive reasoning to write another statement about the graph in Example 4. Then use deductive reasoning to write another statement. 6. SOLUTION Using inductive reasoning: The more strokes it takes for the northern elephant to surface, the deeper it dove. Using deductive reasoning: The northern elephant seal uses fewer strokes to surface from 190 meters then from 410 meters.