1. **State the problem:** We want to find a system of inequalities that models the minimum population $P$ (in thousands) of crickets over time $t$ (in months). 2. **Understand the model:** The population multiplies by 10 every month starting from at least 5 thousand. So the population grows exponentially as $P \geq 5 \times 10^t$. 3. **Stopping condition:** When the population reaches 1 million (which is 1000 thousand), measures stop the increase, so $P \leq 1000$. 4. **Write the system of inequalities:** $$ \begin{cases} P \geq 5 \times 10^t \\ P \leq 1000 \end{cases} $$ 5. **Explanation:** - The first inequality shows the minimum population grows by a factor of 10 each month starting at 5. - The second inequality caps the population at 1000 thousand (1 million). 6. **Check other options:** - $P \geq 10(5)^t$ is incorrect because the base should be 10, not 5. - $P \geq 5(10)t$ is incorrect because it implies linear growth, not exponential. - $P \leq 1000t$ is incorrect because the cap is a fixed number, not increasing with time. **Final answer:** $$ \boxed{\begin{cases} P \geq 5 \times 10^t \\ P \leq 1000 \end{cases}} $$