To solve the inequality $|4c + 5| > 7$, let's proceed step by step:

---

Step 1: Understand the absolute value inequality

For an inequality of the form $|A| > B$, where $B > 0$, we split it into two cases: $A > B \quad \text{or} \quad A < -B.$

Here $A = 4c + 5$ and $B = 7$.

---

Step 2: Split the inequality

We now write: $4c + 5 > 7 \quad \text{or} \quad 4c + 5 < -7.$

---

Step 3: Solve each inequality

Case 1: $4c + 5 > 7$

1. Subtract 5 from both sides: $4c > 2.$
2. Divide by 4: $c > \frac{2}{4}.$
3. Simplify: $c > \frac{1}{2}.$

Case 2: $4c + 5 < -7$

1. Subtract 5 from both sides: $4c < -12.$
2. Divide by 4: $c < \frac{-12}{4}.$
3. Simplify: $c < -3.$

---

Step 4: Combine the solutions

The solutions are: $c > \frac{1}{2} \quad \text{or} \quad c < -3.$

---

Step 5: Express the solution in interval notation

The solution is: $c \in (-\infty, -3) \cup \left(\frac{1}{2}, \infty\right).$

---

Final Answer:

$c \in (-\infty, -3) \cup \left(\frac{1}{2}, \infty\right).$

Would you like a graph or further clarification? Let me know!

---

Related Questions:

1. Solve $|3x - 4| \leq 6$.
2. Solve $|2c + 1| > 3$ and express the solution graphically.
3. Solve $|5x - 2| = 8$.
4. How do you solve $|x| > k$ in general?
5. What are the steps to solve $|A| \leq B$ inequalities?

Tip:

For absolute value inequalities, always remember to split the inequality into two cases—one for the positive and one for the negative side of the expression inside the absolute value.