1. **State the problem:** We are asked to fill in blanks and evaluate expressions involving directed numbers (positive and negative integers), absolute values, and inequalities. 2. **Recall important rules:** - Division by zero is undefined. - The quotient of two integers with the same sign is positive. - The quotient of two integers with different signs is negative. - Absolute value $|a|$ is always non-negative. - Negation of a negative number changes its sign. - Inequalities compare values using $>$ or $<$. --- ### SECTION I: Fill in the blanks 1. $10 \div 0$ is undefined because division by zero is not allowed. 2. The quotient of two integers with the same sign is positive. 3. $0 \div 100 = 0$ because zero divided by any nonzero number is zero. 4. The quotient of two integers with different signs is negative. 5. $- | -5 | = -5$ because $| -5 | = 5$ and negating it gives $-5$. 6. $- | 2 | = -2$ because $|2|=2$ and negating it gives $-2$. 7. $- (-10) = 10$ because negating a negative number makes it positive. 8. Insert $>$ or $<$: a) $-4 < -2$ because $-4$ is less than $-2$. b) $5 > -1$ because $5$ is greater than $-1$. c) $0 > -2$ because $0$ is greater than $-2$. d) $-7 < 2$ because $-7$ is less than $2$. --- ### SECTION II: Evaluate each expression 1. $12 + (-2)(-3)(42)$ Calculate the product first: $(-2)(-3) = 6$ $6 \times 42 = 252$ Then add: $12 + 252 = 264$ 2. $-2(-3)^4 - (-3)$ Calculate $(-3)^4$: $(-3)^4 = (-3) \times (-3) \times (-3) \times (-3) = 81$ Multiply: $-2 \times 81 = -162$ Subtract $-(-3) = +3$: $-162 + 3 = -159$ 3. $18 \div 3(-2)$ Calculate denominator: $3 \times (-2) = -6$ Divide: $18 \div (-6) = -3$ 4. $-3^2 - (-3)^2$ Calculate powers: $-3^2 = -(3^2) = -9$ $(-3)^2 = 9$ Subtract: $-9 - 9 = -18$ 5. $-2 + 2(-5 + 3 \times 4)$ Calculate inside parentheses: $3 \times 4 = 12$ $-5 + 12 = 7$ Multiply: $2 \times 7 = 14$ Add: $-2 + 14 = 12$ 6. $| -4(3) |$ Calculate inside absolute value: $-4 \times 3 = -12$ Absolute value: $| -12 | = 12$ 7. $4 - 2 | -5 - 2 |$ Calculate inside absolute value: $-5 - 2 = -7$ Absolute value: $| -7 | = 7$ Multiply: $2 \times 7 = 14$ Subtract: $4 - 14 = -10$ --- **Final answers:** SECTION I: 1) undefined 2) positive 3) 0 4) negative 5) -5 6) -2 7) 10 8a) < 8b) > 8c) > 8d) < SECTION II: 1) 264 2) -159 3) -3 4) -18 5) 12 6) 12 7) -10