[size=150]Suppose x ∈ Z. If 7x+9 is even, then x is odd.[br][br]Proof. (Contrapositive) Suppose x is not odd.[br][br]Thus x is even, so x = 2a for some integer a.[br][br]Then 7x+9 = 7(2a)+9 = 14a+8+1 = 2(7a+4)+1.[br][br]Therefore 7x+9 = 2b +1, where b is the integer 7a+4.[br][br]Consequently 7x+9 is odd[br][br]Therefore 7x+9 is not even[/size]

[size=150]The given proof uses the method of proof by contrapositive to show that if 7x+9 is even, then x is odd. Let's break down the proof step by step:[br][br]1. Suppose x is not odd. This means x is even, so we can write x = 2a for some integer a.[br][br]2. Substitute the value of x into the expression 7x+9 to obtain 7(2a)+9 = 14a+8+1.[br][br]3. Rearrange the terms to get 14a+8+1 = 2(7a+4)+1.[br][br]4. We can see that 7a+4 is an integer, so let's represent it as b. Then the expression becomes 2b+1.[br][br]5. Therefore, we have shown that 7x+9 can be written as 2b+1, where b is an integer (specifically, b = 7a+4).[br][br]6. From step 5, we can conclude that 7x+9 is odd because it takes the form of 2b+1, where b is an integer.[br][br]7. Consequently, if 7x+9 is odd, it means that 7x+9 is not even.[br][br]By proving the contrapositive, we have established that if 7x+9 is even, then x is odd.[/size]