The problem asks you to compare $-\sqrt{7}$ and $-3.12345\ldots$ and justify your reasoning.

Let's break it down:

1. Estimate $-\sqrt{7}$: $\sqrt{7} \approx 2.64575$ Therefore: $-\sqrt{7} \approx -2.64575$

2. Comparing $-3.12345\ldots$ and $-\sqrt{7}$:

   • We know that $-3.12345\ldots$ is smaller than $-2.64575$ because $-3.12345$ is further left on the number line than $-2.64575$.
   • Thus, $-3.12345 < -\sqrt{7}$.

Justifying the reasoning:

• The square root of 7 is between 2 and 3, meaning $-\sqrt{7}$ is between $-3$ and $-2$.
• Since $-3.12345$ is less than $-3$, it is less than $-\sqrt{7}$.

Filling in the blanks:

1. Blank 1: $-\sqrt{7}$ is between $-3$ and $-2$.
2. Blank 2: Since $-3.12345$ is greater than $-3$, it is less than $-\sqrt{7}$.

Do you need further clarification, or do you have any questions?

Related questions:

1. How do you approximate square roots without a calculator?
2. What does it mean for a number to be greater or less than another on a number line?
3. How does comparing negative numbers differ from comparing positive numbers?
4. What are other ways to estimate irrational numbers like $\sqrt{7}$?
5. Can we compare numbers using fractions instead of decimals?

Tip: When comparing negative numbers, remember that numbers closer to zero are always greater than those further from zero!