Solve 3|x+1| ≥-6. write your solution in interval notation

The inequality to solve is:

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

### Step 1: Isolate the absolute value term
Divide both sides of the inequality by 3 to simplify:

\[
|x+1| \geq \frac{-6}{3}
\]

\[
|x+1| \geq -2
\]

### Step 2: Interpret the inequality
Since the absolute value of any expression is always non-negative (i.e., \(\geq 0\)), the inequality \(|x+1| \geq -2\) is always true for all real numbers. This is because the absolute value of any real number is always greater than or equal to any negative number.

### Step 3: Conclusion
The inequality holds for all real numbers. Therefore, the solution in interval notation is:

\[
(-\infty, \infty)
\]

Let me know if you'd like further details or have any questions!

Here are 5 related questions you might find helpful:
1. How do you solve absolute value inequalities when the right-hand side is positive?
2. What is the difference between \(|x+1| > a\) and \(|x+1| \geq a\)?
3. Can absolute value inequalities have no solution? If so, when?
4. How do you graph the solution to an absolute value inequality?
5. How would the solution change if the inequality were \(|x+1| \leq -6\)?

**Tip:** Always check if an absolute value inequality has a negative number on the right-hand side. If so, the inequality may always hold or have no solution, depending on the inequality sign.

Solve 3|x+1| ≥ -6. Write your solution in interval notation.

Learn how to solve the absolute value inequality 3|x+1| ≥ -6. This problem explores absolute value properties and provides a step-by-step solution, showing that the inequality holds for all real numbers. Suitable for students in Grades 9-10.

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