To solve the inequality $-3|x + 3| > -2$, follow these steps:

Step 1: Isolate the absolute value

First, divide both sides of the inequality by $-3$. Remember to flip the inequality sign since you are dividing by a negative number:

$|x + 3| < \frac{-2}{-3}$

This simplifies to:

$|x + 3| < \frac{2}{3}$

Step 2: Solve the inequality

The expression $|x + 3| < \frac{2}{3}$ implies that the quantity inside the absolute value is within $\frac{2}{3}$ units of 0. Therefore, you can write two inequalities:

$-\frac{2}{3} < x + 3 < \frac{2}{3}$

Step 3: Isolate $x$ in the compound inequality

Subtract 3 from all parts of the inequality:

$-\frac{2}{3} - 3 < x < \frac{2}{3} - 3$

Now, simplify both sides:

$-\frac{2}{3} - \frac{9}{3} < x < \frac{2}{3} - \frac{9}{3}$

$-\frac{11}{3} < x < -\frac{7}{3}$

Step 4: Write the solution in interval notation

The solution to the inequality is:

$\left(-\frac{11}{3}, -\frac{7}{3}\right)$

This interval represents all $x$ values that satisfy the inequality.

Let me know if you need any more details!

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Here are five related questions you might find useful:

1. How do you solve inequalities involving absolute values with a greater than sign?
2. What happens if the inequality was $-3|x + 3| \geq -2$?
3. How can you graph absolute value inequalities?
4. What is the interpretation of an absolute value inequality on a number line?
5. How do you solve compound inequalities involving absolute values?

Tip: Always flip the inequality sign when dividing or multiplying both sides by a negative number, especially when dealing with absolute values.