Final answer:
To prove that x is even if and only if 3x + 5 is odd, you can use a combination of algebra and properties of even and odd functions.
Step-by-step explanation:
To prove that x is even if and only if 3x + 5 is odd, we can use a combination of algebra and properties of even and odd functions.
First, let's assume x is even. This means that x can be written as 2n, where n is an integer.
Substituting this value of x into the expression 3x + 5, we get 3(2n) + 5 = 6n + 5. Since 5 is an odd number and 6n is always even, the sum of an even number and an odd number is always odd. Therefore, 3x + 5 is odd when x is even.
On the other hand, let's assume that 3x + 5 is odd. This means that 3x + 5 can be written as 2m + 1, where m is an integer.
Simplifying this expression, we get 3x + 5 = 2m + 1. Rearranging the terms, we have 3x = 2m - 4. Since 2m - 4 is always even and 3 multiplied by any number is always odd, the only way for the equation to hold is if x is even.
Therefore, x is even if and only if 3x + 5 is odd.