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.