To solve the compound inequality $-7 < 4x + 1 < 7$, we need to break it into two parts and solve for $x$.

Step 1: Solve the left part

$-7 < 4x + 1$ Subtract 1 from both sides: $-7 - 1 < 4x$ $-8 < 4x$ Divide both sides by 4: $\frac{-8}{4} < x$ $-2 < x$

Step 2: Solve the right part

$4x + 1 < 7$ Subtract 1 from both sides: $4x < 7 - 1$ $4x < 6$ Divide both sides by 4: $x < \frac{6}{4}$ $x < 1.5$

Step 3: Combine the results

We now have the solution: $-2 < x < 1.5$ In interval notation, the solution set is: $(-2, 1.5)$

Would you like more details or have any questions?

Here are 5 related questions you can explore:

1. How do you solve inequalities involving absolute values?
2. What is the difference between "and" and "or" compound inequalities?
3. How do you graph the solution of a compound inequality on a number line?
4. Can compound inequalities have no solution? How would you know?
5. What happens if you multiply or divide an inequality by a negative number?

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