We are given the inequality:

$-4|y + 4| < -46$

Step 1: Divide both sides by -4

Since we are dividing by a negative number, we must reverse the inequality.

$|y + 4| > \frac{46}{4}$

Simplify $\frac{46}{4}$:

$|y + 4| > 11.5$

Step 2: Break down the absolute value inequality

Recall that for an absolute value inequality $|A| > B$, it translates to two separate inequalities:

$A > B \quad \text{or} \quad A < -B$

In our case, $A = y + 4$ and $B = 11.5$, so we have:

$y + 4 > 11.5 \quad \text{or} \quad y + 4 < -11.5$

Step 3: Solve each inequality

1. For $y + 4 > 11.5$:

$y > 11.5 - 4$ $y > 7.5$

2. For $y + 4 < -11.5$:

$y < -11.5 - 4$ $y < -15.5$

Final Solution:

The solution to the inequality is:

$y < -15.5 \quad \text{or} \quad y > 7.5$

This is a compound inequality, written as:

$y < -15.5 \quad \text{or} \quad y > 7.5$

Let me know if you'd like further clarification or details!

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Follow-up questions:

1. What happens if we divide an inequality by a negative number? Why is it important to reverse the inequality?
2. How would you solve an absolute value inequality where the expression inside the absolute value is negative?
3. What are the different types of inequalities, and how do you handle them?
4. How would you graph the solution to this compound inequality on a number line?
5. What are some real-world situations where you would encounter absolute value inequalities?

Tip: Always remember to flip the inequality sign when dividing or multiplying by a negative number in inequalities.