To solve the inequality $|3x + 4| < 2$, we need to break it down based on the absolute value definition.

The inequality $|3x + 4| < 2$ means that $3x + 4$ lies between $-2$ and $2$. So, we can write two separate inequalities:

$-2 < 3x + 4 < 2$

Now, we solve this compound inequality by isolating $x$.

Step 1: Subtract 4 from all parts of the inequality

$-2 - 4 < 3x < 2 - 4$ $-6 < 3x < -2$

Step 2: Divide by 3 to isolate $x$

$\frac{-6}{3} < x < \frac{-2}{3}$ $-2 < x < -\frac{2}{3}$

Solution

The solution to the inequality $|3x + 4| < 2$ is: $-2 < x < -\frac{2}{3}$

Would you like a detailed explanation or have any questions?

Here are five related questions to consider:

1. How would the solution change if the inequality were $|3x + 4| \leq 2$?
2. What would the solution be if $|3x + 4| > 2$?
3. How can we graph the solution on a number line?
4. What does the absolute value inequality represent geometrically?
5. How would this inequality change if it were $|3x - 4| < 2$?

Tip: When dealing with absolute value inequalities, split the inequality based on the positive and negative ranges to simplify the solution.