The inequality you provided is \( 4 \leq |b + 6| \).

To solve this inequality, we need to address the absolute value \( |b + 6| \).

### Key Property of Absolute Value Inequality:
For an inequality of the form \( |x| \geq c \), where \( c \geq 0 \), it breaks into two separate inequalities:
\[ x \geq c \quad \text{or} \quad x \leq -c. \]

### Applying this to the given inequality:
Here, \( |b + 6| \geq 4 \), so it breaks into:
\[
b + 6 \geq 4 \quad \text{or} \quad b + 6 \leq -4.
\]

### Solving both inequalities:

1. \( b + 6 \geq 4 \)
   \[
   b \geq 4 - 6
   \]
   \[
   b \geq -2
   \]

2. \( b + 6 \leq -4 \)
   \[
   b \leq -4 - 6
   \]
   \[
   b \leq -10
   \]

### Final solution:
The solution is:
\[
b \geq -2 \quad \text{or} \quad b \leq -10.
\]
In interval notation, this is:
\[
(-\infty, -10] \cup [-2, \infty).
\]

Would you like more details on how the absolute value inequality was split?

### Related questions:
1. How do you solve absolute value inequalities with "<" instead of "≥"?
2. What is the graphical representation of the solution on a number line?
3. How would the inequality change if it were \( 4 > |b + 6| \)?
4. Can inequalities like these be solved using a different method?
5. How does the interval notation express the union of two inequalities?

### Tip:
When solving absolute value inequalities, always remember to split the inequality into two cases—one positive and one negative—to capture all possible solutions.

Learn how to solve the absolute value inequality 4≤|b+6| with a step-by-step breakdown. This problem covers key algebraic concepts, including how to handle absolute value inequalities, and is suitable for students in Grades 9-11.

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