Question

Solve: 3|3x-6|-2≤19
Give your answer as an interval. If no solutions exists - enter No solutions.

Answer 1

To solve the inequality \( 3|3x - 6| - 2 \leq 19 \), we start by isolating the absolute value expression.

First, add 2 to both sides of the inequality:

\[
3|3x - 6| \leq 21
\]

Next, divide both sides by 3:

\[
|3x - 6| \leq 7
\]

Now, we will rewrite this absolute value inequality as a compound inequality:

\[
-7 \leq 3x - 6 \leq 7
\]

We can solve this compound inequality by breaking it into two parts:

1. **First part**:
\[
-7 \leq 3x - 6
\]
Add 6 to both sides:
\[
-1 \leq 3x
\]
Divide by 3:
\[
-\frac{1}{3} \leq x \quad \text{or} \quad x \geq -\frac{1}{3}
\]

2. **Second part**:
\[
3x - 6 \leq 7
\]
Add 6 to both sides:
\[
3x \leq 13
\]
Divide by 3:
\[
x \leq \frac{13}{3}
\]

Combining both parts of the compound inequality, we have:

\[
-\frac{1}{3} \leq x \leq \frac{13}{3}
\]

Thus, the solution in interval notation is:

\[
\left[-\frac{1}{3}, \frac{13}{3}\right]
\]