To solve the inequality \(-3|x + 1| \geq -12\), we can start by dividing both sides by \(-3\). Remember that when dividing by a negative number, the direction of the inequality sign changes:
\[ |x + 1| \leq 4 \]
Next, we can rewrite the absolute value inequality \(|x + 1| \leq 4\) as two separate inequalities:
\[ -4 \leq x + 1 \leq 4 \]
Now we can solve for \(x\) by subtracting \(1\) from all parts of the inequality:
\[ -4 - 1 \leq x \leq 4 - 1 \]
This simplifies to:
\[ -5 \leq x \leq 3 \]
In interval notation, this solution is represented as:
\[ [-5, 3] \]
Thus, the solution to \(-3|x+1| \geq -12\) is:
\[ \boxed{[-5, 3]} \]