https://file.aiquickdraw.com/mathbot/file/1727320956413fp3sqocb.jpg

The equation in the image is:

\[
-4 = |x - 6| + 5
\]

Let's solve this step by step.

### Step 1: Isolate the absolute value
Subtract 5 from both sides of the equation to isolate the absolute value term:

\[
-4 - 5 = |x - 6|
\]

This simplifies to:

\[
-9 = |x - 6|
\]

### Step 2: Analyze the absolute value equation
An absolute value expression like \( |x - 6| \) represents the distance of \( x - 6 \) from zero, and it's always non-negative (i.e., it cannot be less than zero). Therefore:

\[
|x - 6| \geq 0
\]

Since \( -9 \) is negative, there is **no solution** to this equation because the absolute value can never be equal to a negative number.

Thus, the solution set is:

\[
\emptyset \quad \text{(the empty set)}
\]

Would you like any more details or clarifications? 

Here are five questions to expand on this:
1. What is the general rule for solving absolute value equations?
2. How do we handle absolute value inequalities differently from equations?
3. Can an absolute value equation have no solution, and why?
4. What would change if the constant on the right-hand side were positive?
5. How does this equation differ from systems of equations involving absolute values?

**Tip:** Always check the feasibility of absolute value equations by analyzing whether the value on the other side of the equation is non-negative.

Learn how to solve the absolute value equation -4 = |x - 6| + 5 with a step-by-step solution. This problem covers key algebraic concepts, focusing on absolute value and equations. Suitable for students in Grades 9-11.

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