1. **State the problem:** We need to determine which expressions always yield an even value when $x$ is an odd number. 2. **Recall important rules:** - An odd number plus an odd number is even. - An odd number plus an even number is odd. - Odd times even is even. - Odd times odd is odd. - The square of an odd number is always odd. 3. **Analyze each expression:** - Expression 1: $6x$ Since 6 is even, $6 \times x$ is even regardless of $x$ being odd or even. So, $6x$ is always even when $x$ is odd. - Expression 2: $7 + x$ 7 is odd, $x$ is odd, odd + odd = even. So, $7 + x$ is always even when $x$ is odd. - Expression 3: $2x + 3$ $2x$ is even (2 times any integer is even), plus 3 (odd), even + odd = odd. So, $2x + 3$ is odd when $x$ is odd, not always even. - Expression 4: $x^2$ The square of an odd number is odd. So, $x^2$ is odd when $x$ is odd, not even. 4. **Final answer:** The expressions that always have an even value when $x$ is odd are $6x$ and $7 + x$.