https://file.aiquickdraw.com/mathbot/file/17259059686597tx5j04k.jpg

Let's solve the inequality given in the image:

The equation is:
\[
5|x + 3| - 6 \leq -31
\]

### Step 1: Isolate the Absolute Value Expression
To solve this inequality, we first isolate the absolute value term. Start by adding 6 to both sides:
\[
5|x + 3| - 6 + 6 \leq -31 + 6
\]
\[
5|x + 3| \leq -25
\]

### Step 2: Analyze the Inequality
We now have:
\[
5|x + 3| \leq -25
\]

Since the absolute value of any real number is always non-negative (i.e., \( |x| \geq 0 \)), multiplying it by a positive number (5 in this case) will also give a non-negative result. Therefore, it is **impossible** for \( 5|x + 3| \) to be less than or equal to -25.

### Conclusion
There are **no solutions** to this inequality because it is impossible for a non-negative value to satisfy a less-than-or-equal-to condition with a negative number. 

Thus, the correct answer is "No solution."

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Would you like more details or have any other questions?

Here are 5 related questions to expand your knowledge:
1. How do you solve absolute value equations?
2. Can an absolute value expression ever be negative?
3. What would happen if the inequality was \(5|x + 3| \leq 25\) instead?
4. What steps would you take if the inequality was \(5|x + 3| - 6 \geq 31\)?
5. How do you solve compound inequalities involving absolute values?

**Tip:** Always check if the inequality is feasible when working with absolute values—remember, absolute values cannot be negative!

Learn how to solve the absolute value inequality 5|x + 3| - 6 ≤ -31. This step-by-step solution focuses on algebraic concepts such as absolute value properties and inequalities, demonstrating that this inequality has no solution. Suitable for high school students in Grades 9-10.

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