1. **Problem 8:** You subtract two integers and the difference is -8. Find two pairs of integers that work. - First pair: both minuend and subtrahend are negative. - Second pair: minuend is positive and subtrahend is negative. **Step 1:** Recall subtraction formula: $$\text{difference} = \text{minuend} - \text{subtrahend}$$ **Step 2:** For the first pair (both negative), let minuend = $-a$, subtrahend = $-b$ where $a,b > 0$. $$-a - (-b) = -a + b = -8$$ Rearranged: $$b - a = -8$$ Choose $a=10$, then $b = 2$ because $2 - 10 = -8$. So first pair: minuend = $-10$, subtrahend = $-2$. **Step 3:** For the second pair (minuend positive, subtrahend negative), let minuend = $m > 0$, subtrahend = $-n$ where $n > 0$. $$m - (-n) = m + n = -8$$ Since $m + n$ is sum of two positive numbers, it cannot be negative. Therefore, no solution exists for the difference to be $-8$ if minuend is positive and subtrahend is negative. **Answer for 8:** - First pair: $(-10, -2)$ - Second pair: No solution --- 2. **Problem 9:** You multiply a number by $-4$ and the product is positive. Find two numbers you could have multiplied by $-4$ and two you could not. **Step 1:** Recall multiplication sign rules: - Negative times negative = positive - Negative times positive = negative **Step 2:** Since product is positive and one factor is $-4$ (negative), the other number must be negative. **Step 3:** Examples of numbers you could multiply by $-4$ to get positive product: - $-1$ because $-4 \times -1 = 4$ (positive) - $-3$ because $-4 \times -3 = 12$ (positive) **Step 4:** Examples of numbers you could not multiply by $-4$ to get positive product: - $2$ because $-4 \times 2 = -8$ (negative) - $5$ because $-4 \times 5 = -20$ (negative) **Answer for 9:** - Numbers you could multiply by: $-1$, $-3$ - Numbers you could not multiply by: $2$, $5$