Let's solve the equation \(|-10x - 3x| - 1 = 19\).

### Step 1: Simplify the expression inside the absolute value
Combine the like terms inside the absolute value:

\[
|-10x - 3x| = |-13x|
\]

So the equation becomes:

\[
|-13x| - 1 = 19
\]

### Step 2: Isolate the absolute value
Add 1 to both sides of the equation:

\[
|-13x| = 20
\]

### Step 3: Solve the absolute value equation
The equation \(|-13x| = 20\) can be broken down into two possible cases:

1. **Case 1:**
   \[
   -13x = 20
   \]
   Solve for \(x\):
   \[
   x = -\frac{20}{13}
   \]

2. **Case 2:**
   \[
   -13x = -20
   \]
   Solve for \(x\):
   \[
   x = \frac{20}{13}
   \]

### Final Answer:
The solutions are:
\[
x = -\frac{20}{13} \quad \text{or} \quad x = \frac{20}{13}
\]

Would you like more details or have any questions? 

Here are some related questions you might find interesting:
1. How do you solve equations with multiple absolute values?
2. What happens if the right-hand side of an absolute value equation is negative?
3. How can you graph an absolute value function?
4. What is the significance of combining like terms in algebra?
5. How do you check if your solutions are correct in an absolute value equation?

**Tip:** When solving absolute value equations, remember to consider both the positive and negative cases to ensure all possible solutions are found.

Learn how to solve the absolute value equation |-10x - 3x| - 1 = 19 with a detailed step-by-step solution. This problem involves absolute value concepts and algebraic equations. Suitable for students in Grades 9-12.

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