Certainly! Let's solve the inequality step-by-step.
Given inequality:
[tex]\[ 3|x - 6| \leq 9 \][/tex]
Step 1: Isolate the absolute value expression
First, divide both sides of the inequality by 3:
[tex]\[ |x - 6| \leq 3 \][/tex]
Step 2: Understand the absolute value inequality
The absolute value inequality [tex]\( |x - 6| \leq 3 \)[/tex] can be interpreted as:
[tex]\[ -3 \leq x - 6 \leq 3 \][/tex]
Step 3: Solve the compound inequality
We can break this compound inequality down into two parts to solve for [tex]\( x \)[/tex]:
1. Solve the left part:
[tex]\[ -3 \leq x - 6 \][/tex]
Add 6 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ -3 + 6 \leq x \][/tex]
[tex]\[ 3 \leq x \][/tex]
2. Solve the right part:
[tex]\[ x - 6 \leq 3 \][/tex]
Add 6 to both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x - 6 + 6 \leq 3 + 6 \][/tex]
[tex]\[ x \leq 9 \][/tex]
Step 4: Combine the results
Combining [tex]\( 3 \leq x \)[/tex] and [tex]\( x \leq 9 \)[/tex] gives us the final interval:
[tex]\[ 3 \leq x \leq 9 \][/tex]
So, the solution to the inequality [tex]\( 3|x - 6| \leq 9 \)[/tex] is:
[tex]\[ 3 \leq x \leq 9 \][/tex]
This means that any [tex]\( x \)[/tex] between 3 and 9, inclusive, satisfies the inequality.
