1. **State the problem:** We need to determine which pair of expressions are equivalent by simplifying each and comparing. 2. **Check pair 1: $4 + 12x$ and $4(3x + 1)$** Simplify $4(3x + 1)$: $$4(3x + 1) = 4 \times 3x + 4 \times 1 = 12x + 4$$ This matches $4 + 12x$ exactly, so they are equivalent. 3. **Check pair 2: $9x$ and $2 \times 4(x + 1) - 1$** Simplify $2 \times 4(x + 1) - 1$: $$2 \times 4(x + 1) - 1 = 8(x + 1) - 1 = 8x + 8 - 1 = 8x + 7$$ Compare to $9x$; since $8x + 7 \neq 9x$, they are not equivalent. 4. **Check pair 3: $5(x - 1) + (1 - x)$ and $6x - 4$** Simplify $5(x - 1) + (1 - x)$: $$5x - 5 + 1 - x = (5x - x) + (-5 + 1) = 4x - 4$$ Compare to $6x - 4$; since $4x - 4 \neq 6x - 4$, they are not equivalent. 5. **Check pair 4: $2(x + 1) + (x - 1)$ and $3x + 1$** Simplify $2(x + 1) + (x - 1)$: $$2x + 2 + x - 1 = (2x + x) + (2 - 1) = 3x + 1$$ They are equivalent. 6. **Check pair 5: $7x + \frac{1}{3} - 5x + 1 \frac{1}{3}$ and $2x + 1 \frac{2}{3}$** First, convert mixed fractions to improper fractions: $$1 \frac{1}{3} = \frac{4}{3}, \quad 1 \frac{2}{3} = \frac{5}{3}$$ Simplify left side: $$7x + \frac{1}{3} - 5x + \frac{4}{3} = (7x - 5x) + \left(\frac{1}{3} + \frac{4}{3}\right) = 2x + \frac{5}{3}$$ Right side is $2x + \frac{5}{3}$, so they are equivalent. **Final answer:** The equivalent pairs are: - $4 + 12x$ and $4(3x + 1)$ - $2(x + 1) + (x - 1)$ and $3x + 1$ - $7x + \frac{1}{3} - 5x + 1 \frac{1}{3}$ and $2x + 1 \frac{2}{3}$ --- **Next, interpret the equation $368 = 8g$:** This equation means the total number 368 equals 8 times $g$. If $g$ represents the number of days, then $8g$ is the total number of students absent over $g$ days if 8 students are absent each day. So, $368 = 8g$ models the situation where 368 students are absent in total, with 8 absent each day, and $g$ is the number of days. This matches situation (A): "A school has 368 students. Each day 8 students are absent. How many students are absent at the end of one school week?" where $g$ would be the number of days in the school week. --- **Summary:** - Equivalent pairs: 1, 4, and 5. - Equation $368 = 8g$ corresponds to situation (A).