12x+7< -11 \quad \maroonC{\text{ OR }} \quad 5x-8>40

We are given two inequalities joined by the logical operator "OR":

\[
12x + 7 < -11 \quad \text{OR} \quad 5x - 8 > 40
\]

We will solve each inequality separately.

### Inequality 1: \( 12x + 7 < -11 \)

1. Subtract 7 from both sides:
   \[
   12x < -11 - 7
   \]
   \[
   12x < -18
   \]

2. Divide both sides by 12:
   \[
   x < \frac{-18}{12}
   \]
   \[
   x < -\frac{3}{2}
   \]
   \[
   x < -1.5
   \]

### Inequality 2: \( 5x - 8 > 40 \)

1. Add 8 to both sides:
   \[
   5x > 40 + 8
   \]
   \[
   5x > 48
   \]

2. Divide both sides by 5:
   \[
   x > \frac{48}{5}
   \]
   \[
   x > 9.6
   \]

### Final Answer:

The solution to the inequality is:
\[
x < -1.5 \quad \text{OR} \quad x > 9.6
\]

Would you like further clarification on any step?

---

**Related Questions:**

1. What happens if we use "AND" instead of "OR" in inequalities?
2. How do we solve compound inequalities involving "AND" and "OR"?
3. Can you explain the difference between strict and non-strict inequalities?
4. How do we graph the solution of a compound inequality?
5. What happens if a variable appears in both parts of a compound inequality?

**Tip:** Always double-check the operation performed when solving inequalities, especially when dividing by negative numbers (which requires flipping the inequality sign).

Learn how to solve compound inequalities like 12x + 7 < -11 OR 5x - 8 > 40 step by step. This detailed guide breaks down each inequality, demonstrates the solution process, and explains the final answer: x < -1.5 OR x > 9.6. Perfect for students in Grades 9-11 studying linear inequalities and compound inequalities.

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