https://file.aiquickdraw.com/mathbot/file/1728360384628pq9atxge.jpg

Let's solve the inequality given in the image:

The inequality shown is:

\[
8|x + 4| + 10 < 2
\]

### Step-by-step solution:

1. **Isolate the absolute value expression**:
   
   Subtract 10 from both sides:

   \[
   8|x + 4| < 2 - 10
   \]
   
   \[
   8|x + 4| < -8
   \]

2. **Interpret the result**:
   
   Since the absolute value of any number is always non-negative (i.e., \( |x| \geq 0 \) for all \(x\)), multiplying the absolute value by 8 gives a non-negative result.

   However, here we have:

   \[
   8|x + 4| < -8
   \]

   This inequality states that a non-negative number must be less than \(-8\), which is impossible because absolute values cannot be negative.

### Conclusion:

There is **no solution** to this inequality because the expression \( 8|x + 4| \) can never be less than \(-8\).

---

Would you like to ask any further details or clarifications?

Here are some related questions that you might want to explore:
1. What is the difference between solving equations and inequalities involving absolute values?
2. Can an inequality ever have no solution? If yes, why?
3. How do absolute value properties affect inequalities?
4. What are other methods to approach inequalities with absolute values?
5. How would the inequality change if the constant on the right-hand side were positive instead of negative?

**Tip**: When solving inequalities with absolute values, always consider the possibility of the inequality being impossible if it leads to a negative comparison with an absolute value term.

Learn how to solve the absolute value inequality 8|x + 4| + 10 < 2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the absolute value theorem and properties of inequalities. Suitable for students in Grades 9-10.

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