To prove that $x$ is an odd integer if and only if $5x - 1$ is an even integer, we need to demonstrate both implications:

1. If $x$ is an odd integer, then $5x - 1$ is an even integer.

Assume $x$ is an odd integer. By definition, an odd integer can be expressed as: $x = 2k + 1$ for some integer $k$.

Now, substitute this into the expression $5x - 1$: $5x - 1 = 5(2k + 1) - 1$ Expanding this: $5x - 1 = 10k + 5 - 1 = 10k + 4$ Notice that $10k + 4$ can be factored as: $5x - 1 = 2(5k + 2)$ Since $5k + 2$ is an integer, $5x - 1$ is twice some integer, which means it is an even integer.

Thus, if $x$ is an odd integer, $5x - 1$ is indeed an even integer.

2. If $5x - 1$ is an even integer, then $x$ is an odd integer.

Assume $5x - 1$ is an even integer. This means: $5x - 1 = 2m$ for some integer $m$.

To solve for $x$, add 1 to both sides: $5x = 2m + 1$ Now divide by 5: $x = \frac{2m + 1}{5}$ Here, $2m + 1$ is an odd integer (since the sum of an even and an odd number is odd), and an odd integer divided by 5 will result in another integer only if $x$ is odd.

This implies that $x$ must be of the form $x = 2k + 1$ for some integer $k$, which means $x$ is an odd integer.

Thus, if $5x - 1$ is an even integer, $x$ must be an odd integer.

Conclusion

We have shown both directions:

1. If $x$ is an odd integer, then $5x - 1$ is an even integer.
2. If $5x - 1$ is an even integer, then $x$ is an odd integer.

Therefore, $x$ is an odd integer if and only if $5x - 1$ is an even integer.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What happens if $x$ is an even integer in the expression $5x - 1$?
2. How would the proof change if the expression were $5x + 1$ instead of $5x - 1$?
3. Can you prove that if $x$ is even, then $5x - 1$ is odd?
4. How does this proof apply to other linear functions of $x$?
5. Can you find another similar statement involving odd/even integers and a different linear combination?

Tip: Remember that even and odd integers have a predictable behavior when added, subtracted, or multiplied, which is helpful in proving statements involving parity.