solve -3|x + 3| > -2. give your answer in interval notation. enter one if the inequality does not have a solution

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.

Learn how to solve the absolute value inequality -3|x + 3| > -2 step-by-step with detailed explanations. This problem involves inequalities and absolute values, suitable for high school students. Explore solutions and clear explanations for -3|x + 3| > -2.

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