To solve the equation \( \left\vert -4x - 4 \right\vert = 22 \), we need to consider the definition of absolute value, which splits the equation into two cases:

### Case 1: 
\[ -4x - 4 = 22 \]

### Case 2: 
\[ -4x - 4 = -22 \]

Let's solve each case separately.

### Case 1:
\[
-4x - 4 = 22
\]
Add 4 to both sides:
\[
-4x = 26
\]
Divide both sides by -4:
\[
x = -\frac{26}{4} = -\frac{13}{2} = -6.5
\]

### Case 2:
\[
-4x - 4 = -22
\]
Add 4 to both sides:
\[
-4x = -18
\]
Divide both sides by -4:
\[
x = \frac{18}{4} = \frac{9}{2} = 4.5
\]

### Final Answer:
The solutions to the equation are:
\[
x = -6.5 \quad \text{or} \quad x = 4.5
\]

Would you like more details or have any questions? Here are some related problems:

1. Solve \( \left\vert 3x + 5 \right\vert = 17 \).
2. Solve \( \left\vert 2x - 7 \right\vert = 15 \).
3. Find the solutions to \( \left\vert x + 2 \right\vert = 10 \).
4. Solve the equation \( 5\left\vert x - 1 \right\vert = 35 \).
5. What are the solutions to \( \left\vert x^2 - 4x \right\vert = 16 \)?

**Tip:** Remember, when solving absolute value equations, always consider both the positive and negative scenarios that could satisfy the equation.

Learn how to solve the absolute value equation |x + 1| = 22 with a step-by-step solution. This problem covers absolute value concepts and linear equations, suitable for high school students in Grades 9-12.

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