To solve the inequality \(-3|x+1| \geq -12\), we can start by dividing both sides by \(-3\). Remember that when we divide an inequality by a negative number, we must reverse the inequality sign:
\[ |x + 1| \leq 4 \]
Next, we interpret the absolute value inequality \( |x + 1| \leq 4 \). This means that the expression inside the absolute value lies between \(-4\) and \(4\):
\[ -4 \leq x + 1 \leq 4 \]
We can solve this compound inequality by separating it into two inequalities:
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From \( -4 \leq x + 1 \): \[ -4 - 1 \leq x \implies -5 \leq x \]
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From \( x + 1 \leq 4 \): \[ x \leq 4 - 1 \implies x \leq 3 \]
Combining the results, we get:
\[ -5 \leq x \leq 3 \]
In interval notation, the solution is:
\[ [-5, 3] \]