The inequality presented in the image is:

$|9x + 7| < 4$

To solve this absolute value inequality, we follow these steps:

Step 1: Remove the absolute value by setting up two inequalities

The expression $|A| < B$ means that $-B < A < B$. Therefore, we can write:

$-4 < 9x + 7 < 4$

Step 2: Solve the compound inequality

We now solve the inequality in two parts.

Part 1: Solve $-4 < 9x + 7$

• Subtract 7 from both sides: $-4 - 7 < 9x$ $-11 < 9x$
• Divide by 9: $\frac{-11}{9} < x$ $x > \frac{-11}{9}$

Part 2: Solve $9x + 7 < 4$

• Subtract 7 from both sides: $9x < 4 - 7$ $9x < -3$
• Divide by 9: $x < \frac{-3}{9}$ $x < \frac{-1}{3}$

Step 3: Combine the two results

We have: $\frac{-11}{9} < x < \frac{-1}{3}$

Thus, the solution is: $x \in \left( \frac{-11}{9}, \frac{-1}{3} \right)$

Would you like a more detailed explanation of any step or have any other questions?

Here are 5 related questions:

1. How do you solve absolute value inequalities when the sign is "greater than" (>)?
2. What happens if you divide an inequality by a negative number?
3. How do you graph inequalities on a number line?
4. What are the differences between solving equations and inequalities?
5. How would this solution change if the inequality were $|9x + 7| \leq 4$?

Tip: When dealing with absolute values in inequalities, always remember that they split into two inequalities, one positive and one negative.