Question

Find the solution to −3|x+1|≥−12 . Provide your answer using interval notation.(1 point)
The solution to −3|x+1|≥−12
is [
].

Answer 1

To solve the inequality \(-3|x+1| \geq -12\), we can first divide both sides by \(-3\). Remember that dividing by a negative number flips the inequality sign.

\[
|x+1| \leq 4
\]

The absolute value inequality \(|x+1| \leq 4\) means that the expression inside the absolute value, \(x + 1\), must lie within the interval \([-4, 4]\). We can express this as two separate inequalities:

\[
-4 \leq x + 1 \leq 4
\]

Now, we can solve for \(x\):

1. From \(-4 \leq x + 1\):

\[
-4 - 1 \leq x \implies -5 \leq x
\]

2. From \(x + 1 \leq 4\):

\[
x \leq 4 - 1 \implies x \leq 3
\]

Now we have:

\[
-5 \leq x \leq 3
\]

In interval notation, this is written as:

\[
[-5, 3]
\]

Thus, the solution to the inequality \(-3|x+1| \geq -12\) is:

\[
\boxed{[-5, 3]}
\]