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The inequality given is:

\[
|x + 1| \geq 6
\]

### Steps to solve:

1. **Break down the absolute value inequality**:
   The expression \( |x + 1| \geq 6 \) implies two cases:
   
   - \( x + 1 \geq 6 \)
   - \( x + 1 \leq -6 \)

2. **Solve each case**:
   - **Case 1**: 
     \[
     x + 1 \geq 6 \implies x \geq 5
     \]
   
   - **Case 2**:
     \[
     x + 1 \leq -6 \implies x \leq -7
     \]

3. **Combine the solutions**:
   The inequality holds true for \( x \geq 5 \) **or** \( x \leq -7 \).

### Graphing the solution:
- Draw a number line with key points at \( x = -7 \) and \( x = 5 \).
- Shade the region where \( x \leq -7 \) (i.e., all values to the left of \( -7 \)) and \( x \geq 5 \) (i.e., all values to the right of \( 5 \)).
- Since the inequality is **greater than or equal to**, include the points \( x = -7 \) and \( x = 5 \) with solid circles at these points.

Would you like me to sketch this graph for you, or provide additional details?

### Related Questions:
1. What is the difference between strict and non-strict inequalities in graphing?
2. How would the graph change if the inequality was \( |x + 1| < 6 \)?
3. How can absolute value inequalities be solved using a graphical method?
4. Can this inequality be solved using interval notation? If yes, how?
5. What is the general form for solving absolute value inequalities?

**Tip**: When solving absolute value inequalities, always remember to consider both the positive and negative cases of the expression inside the absolute value.

Sketch the graph of the given inequality: |x + 1| ≥ 6

Learn how to solve and graph the absolute value inequality |x + 1| ≥ 6. This problem explains the steps involved in breaking down the inequality into two cases and graphing the solution on a number line. Suitable for students in Grades 8-10.

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