The image shows a compound inequality:

$-4 < 3x + 2 \leq 18$

To solve this inequality, we need to isolate $x$ in the middle. Here’s a step-by-step solution:

Step 1: Subtract 2 from all parts of the inequality

$-4 - 2 < 3x + 2 - 2 \leq 18 - 2$ $-6 < 3x \leq 16$

Step 2: Divide all parts by 3 to solve for $x$

$\frac{-6}{3} < x \leq \frac{16}{3}$ $-2 < x \leq \frac{16}{3}$

Solution

The solution to the inequality is: $-2 < x \leq \frac{16}{3}$

This can also be written in interval notation as: $(-2, \frac{16}{3}]$

Would you like further details on any of these steps?

Follow-up Questions:

1. What is a compound inequality, and how is it different from a single inequality?
2. How do we interpret the solution interval $(-2, \frac{16}{3}]$ on a number line?
3. What happens to the inequality if we reverse the direction of the signs?
4. How do we check if a specific value of $x$ satisfies this inequality?
5. How does solving compound inequalities differ from solving equations?

Tip:

When solving inequalities, always pay attention to the direction of the inequality signs, especially when multiplying or dividing by negative numbers.