Sure! Let's solve the inequality [tex]\(\left|\frac{x-2}{3}\right| \leq 3\)[/tex] step by step.
### Step 1: Understand the Absolute Value Inequality
The inequality given is [tex]\(\left|\frac{x-2}{3}\right| \leq 3\)[/tex]. This absolute value inequality states that the expression inside the absolute value, [tex]\(\frac{x-2}{3}\)[/tex], lies within the range of [tex]\(-3\)[/tex] to [tex]\(3\)[/tex].
### Step 2: Remove the Absolute Value
To remove the absolute value, we set up two separate inequalities:
[tex]\[ \frac{x-2}{3} \leq 3 \][/tex]
and
[tex]\[ \frac{x-2}{3} \geq -3 \][/tex]
### Step 3: Solve Each Inequality Separately
#### Solve [tex]\(\frac{x-2}{3} \leq 3\)[/tex]
1. Multiply both sides of the inequality by 3 to clear the denominator:
[tex]\[ \frac{x-2}{3} \leq 3 \][/tex]
[tex]\[ x - 2 \leq 9 \][/tex]
2. Add 2 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ x \leq 11 \][/tex]
#### Solve [tex]\(\frac{x-2}{3} \geq -3\)[/tex]
1. Similarly, multiply both sides by 3:
[tex]\[ \frac{x-2}{3} \geq -3 \][/tex]
[tex]\[ x - 2 \geq -9 \][/tex]
2. Add 2 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[ x \geq -7 \][/tex]
### Step 4: Combine the Results
The solutions from both inequalities give us:
[tex]\[ -7 \leq x \leq 11 \][/tex]
Therefore, the solution to the inequality [tex]\(\left|\frac{x-2}{3}\right| \leq 3\)[/tex] is:
[tex]\[ x \in [-7, 11] \][/tex]
This means that [tex]\(x\)[/tex] can be any value from [tex]\(-7\)[/tex] to [tex]\(11\)[/tex], inclusive.
