To solve the inequality $|4m + 6| > 4$, we proceed as follows:

Step 1: Understand the inequality for absolute value

The inequality $|4m + 6| > 4$ splits into two cases:

1. $4m + 6 > 4$
2. $4m + 6 < -4$

Step 2: Solve each case

Case 1: $4m + 6 > 4$

$4m > 4 - 6 \implies 4m > -2 \implies m > -\frac{1}{2}$

Case 2: $4m + 6 < -4$

$4m < -4 - 6 \implies 4m < -10 \implies m < -\frac{5}{2}$

Step 3: Combine the two cases

The solution is the union of the two regions: $m < -\frac{5}{2} \quad \text{or} \quad m > -\frac{1}{2}$

Step 4: Match with the options

This corresponds to Option D: $\{m < -\frac{5}{2} \, \text{or} \, m > -\frac{1}{2}\}.$

Would you like further clarifications or detailed explanations of the steps?

5 Related Questions:

1. What happens if the inequality was $|4m + 6| \leq 4$?
2. How do you graph the solution set for this inequality on a number line?
3. What are the key differences between solving $|x| > c$ and $|x| < c$?
4. How can you rewrite $|x| > c$ in terms of compound inequalities?
5. How do absolute value inequalities appear in real-world problems?

Tip:

Always remember to split absolute value inequalities into two cases, one for the positive scenario and one for the negative scenario!