Problem 4 Making a Prediction

Sales Sales of backpacks at a nationwide company decreased over a period of six consecutive months. What conjecture can you make about the number of backpacks the company will sell in May?

The points seem to fall on a line. The graph shows the number of sales decreasing by about 500 backpacks each month. By inductive reasoning, you can estimate that the company will sell approximately 8000 backpacks in May.

✓ Got It?

4. 1. What conjecture can you make about backpack sales in June?
   2. Reasoning Is it reasonable to use this graph to make a conjecture about sales in August? Explain.

Problem 5 Finding a Counterexample

What is a counterexample for each conjecture?

1. If the name of a month starts with the letter J, it is a summer month.

   Counterexample: January starts with J and it is a winter month.

2. You can connect any three points to form a triangle.

   Counterexample: If the three points lie on a line, you cannot form a triangle.

   

   

3. When you multiply a number by 2, the product is greater than the original number.

   Think

   What numbers should you guess-and-check?

   Try positive numbers, negative numbers, fractions, and special cases like zero.

   The conjecture is true for positive numbers, but it is false for negative numbers and zero.

   Counterexample: − 4 · 2 = − 8 negative . 4 middle dot 2 equals negative 8  and − 8 ≯ − 4 . negative 8 not greater than negative 4 .

✓ Got It?

5. What is a counterexample for each conjecture?
   1. If a flower is red, it is a rose.
   2. One and only one plane exists through any three points.
   3. When you multiply a number by 3, the product is divisible by 6.