Let's solve the inequality step by step using the addition and multiplication principles, as the problem requires.

The inequality given is:

$11x - 10 < -43$

Step 1: Add 10 to both sides

We want to isolate the term with $x$, so we start by adding 10 to both sides:

$11x - 10 + 10 < -43 + 10$

Simplifying:

$11x < -33$

Step 2: Divide both sides by 11

Now, divide both sides by 11 to solve for $x$:

$x < \frac{-33}{11}$

Simplifying:

$x < -3$

Conclusion:

The solution to the inequality is $x < -3$.

Now, let's look at the choices:

• A: The solution set is $\{ x \mid x \geq \}$
• B: The solution set is $\{ x \mid x < \}$
• C: The solution set is $\{ x \mid x > \}$
• D: The solution set is $\{ x \mid x \leq \}$

The correct answer corresponds to choice B, where $x < -3$.

Would you like more details or have any questions?

Here are five related questions for you to consider:

1. How would the inequality change if the $11x - 10$ were on the right side instead?
2. What happens when multiplying or dividing inequalities by negative numbers?
3. How do you graph the solution $x < -3$ on a number line?
4. Can you solve a similar inequality where the $x$-term has a fractional coefficient?
5. How would the solution change if the inequality was $11x - 10 > -43$?

Tip: Always reverse the inequality sign when multiplying or dividing both sides of an inequality by a negative number!