Question

Determine if the conjecture is true or false. If true, explain your reasoning briefly. If false provide a counterexample.

A. Given that 1/5
`\leq`5,
1/20
`\leq`, 20,
and 1/38
`\leq`38, you conjecture that 1/x
`\leq`x all real numbers x
`\leq` 0

Answer 1

9514 1404 393

Answer:

false

Explanation:

The conjecture shown in this problem statement does not follow from the examples offered. They support the notion that ...

1/x ≤ x . . . . for x ≥ 0 (not x ≤ 0)

There are several possible counterexamples showing the conjecture is FALSE.

1. 1/0 is undefined
2. 1/(-5) > -5 . . . . . . . . a case for x < 0

If the intended domain is x ≥ 0, then the conjecture can also be demonstrated to be false for 0 < x < 1:

1. 1/(1/5) > 1/5