Question

Solve the inequality. Then graph the solution and give interval notation.
-18 ≦ -3x-6<3
0
1
2
3
4
5
-1
-2
-3
-4
-5
Interval notation for the above graph and inequality is

Answer 1

To solve the compound inequality \(-18 \leq -3x - 6 < 3\), we will break it down into two parts:

1. \(-18 \leq -3x - 6\)
2. \(-3x - 6 < 3\)

### Step 1: Solve \(-18 \leq -3x - 6\)

Add 6 to both sides:

\[
-18 + 6 \leq -3x
\]

\[
-12 \leq -3x
\]

Now, divide both sides by -3. Remember to reverse the inequality symbol when dividing by a negative number:

\[
\frac{-12}{-3} \geq x
\]

\[
4 \geq x \quad \text{or} \quad x \leq 4
\]

### Step 2: Solve \(-3x - 6 < 3\)

Add 6 to both sides:

\[
-3x < 3 + 6
\]

\[
-3x < 9
\]

Now, divide both sides by -3, reversing the inequality:

\[
x > \frac{9}{-3}
\]

\[
x > -3
\]

### Step 3: Combine the results

From the two solved parts, we have:

1. \(x \leq 4\)
2. \(x > -3\)

Combining these gives us:

\[
-3 < x \leq 4
\]

### Interval Notation

In interval notation, this is written as:

\[
(-3, 4]
\]

### Graphing the solution

- The graph of \(x > -3\) will be represented with an open circle at -3, indicating that -3 is not included in the solution.
- The graph of \(x \leq 4\) will be represented with a closed circle at 4, indicating that 4 is included in the solution.

### Final Answer

So the interval notation for the solution is:

\[
(-3, 4]
\]