Problem 54 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I made a mistake when I used \(x\) and \(x+2\) to represent two consecutive odd integers, because 2 is even.

Step-by-Step Solution

Answer

The statement does not make sense. Using 'x' and 'x+2' to represent consecutive odd integers is correct because the difference between two consecutive odd numbers is always 2 even though '2' is an even number.

Step 1: Understanding Consecutive Odd Integers

Odd integers are numbers that have a remainder of 1 when divided by 2. Consecutive odd integers follow a specific pattern. For example, the first few odd numbers are 1, 3, 5, 7, and 9. It can be seen from this pattern that each odd number is 2 more than the preceding odd number.

Step 2: Exploring the Indices

Reviewing the statement, 'x' and 'x+2' are used to represent two consecutive odd integers. If 'x' represents an odd integer, then the next consecutive odd integer will be 'x+2'. The reason for adding two instead of one is because adding one to an odd integer always results in an even integer, and we need the next odd integer. So 'x+2' effectively skips the next even number to give the next odd integer.

Step 3: Conclusion

With this understanding, it's clear that using 'x' and 'x+2' to represent two consecutive odd integers is completely reasonable, even though '2' is an even number. The student's mistake lies in misunderstanding the concept of representation of consecutive odd integers.