You are given the inequality \( 12 < -5x - 3 \) or \( -5x - 3 < -23 \). Let's solve each part separately:

### First inequality:
\[ 12 < -5x - 3 \]
1. Add 3 to both sides:
\[
12 + 3 < -5x \quad \Rightarrow \quad 15 < -5x
\]
2. Divide both sides by -5, remembering to reverse the inequality sign when dividing by a negative number:
\[
\frac{15}{-5} > x \quad \Rightarrow \quad -3 > x \quad \text{or} \quad x < -3
\]

### Second inequality:
\[ -5x - 3 < -23 \]
1. Add 3 to both sides:
\[
-5x - 3 + 3 < -23 + 3 \quad \Rightarrow \quad -5x < -20
\]
2. Divide both sides by -5, again reversing the inequality sign:
\[
\frac{-5x}{-5} > \frac{-20}{-5} \quad \Rightarrow \quad x > 4
\]

### Conclusion:
The solution to the inequality is:
\[
x < -3 \quad \text{or} \quad x > 4
\]
This means that \( x \) can either be less than -3 or greater than 4.

Would you like more details or have any questions?

### Related Questions:
1. How do you solve compound inequalities involving "and"?
2. What happens if both inequalities are combined with "and" instead of "or"?
3. How do you graph inequalities on a number line?
4. How does reversing the inequality sign work when dividing or multiplying by negatives?
5. Can you show how these inequalities can be written in interval notation?

### Tip:
Always remember to reverse the inequality sign when multiplying or dividing by a negative number in inequalities!

Learn how to solve the compound inequality 12 < −5x − 3 or −5x − 3 < −23 step by step. This guide explains key algebraic concepts such as reversing inequality signs when dividing by a negative number, and is suitable for students in Grades 8-10.

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