We begin by solving the equation:

\[
|7x + 6| + 9 = 1
\]

### Step 1: Isolate the absolute value expression
First, subtract 9 from both sides to isolate the absolute value term:

\[
|7x + 6| = 1 - 9
\]

\[
|7x + 6| = -8
\]

### Step 2: Analyze the result
We have:

\[
|7x + 6| = -8
\]

Since the absolute value of any expression is always non-negative (i.e., it cannot be less than zero), there is **no solution** because there is no value of \( x \) that can make the absolute value equal to a negative number.

### Final Answer:
The solution set is the **empty set** (∅), meaning there is no real solution.

Would you like more details or have any questions about this solution? Here are some related questions to consider:

1. How do we solve absolute value equations in general?
2. What would happen if the right-hand side was a positive number?
3. How can you solve absolute inequalities like \( |7x + 6| < 8 \)?
4. What happens when an absolute value equation has no solution?
5. Can we graph the function \( |7x + 6| + 9 \) to verify the solution visually?

**Tip**: Always check whether the number on the right-hand side of an absolute value equation is non-negative, as absolute values can't be negative.

Learn how to solve the absolute value equation |7x + 6| + 9 = 1, exploring algebraic concepts such as isolating absolute value expressions and understanding when an equation has no solution. Suitable for students in Grades 8-10.

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