To solve the inequality [tex]\( |2x - 8| \leq 6 \)[/tex], you need to break it into two separate inequalities and solve them individually. The absolute value inequality [tex]\( |A| \leq B \)[/tex] can be rewritten as:
[tex]\[ -B \leq A \leq B \][/tex]
In our specific problem, [tex]\( A = 2x - 8 \)[/tex] and [tex]\( B = 6 \)[/tex], so we have:
[tex]\[ -6 \leq 2x - 8 \leq 6 \][/tex]
We'll solve this compound inequality in two steps.
### Step 1: Solve [tex]\( -6 \leq 2x - 8 \)[/tex]
1. Add 8 to both sides:
[tex]\[ -6 + 8 \leq 2x \][/tex]
[tex]\[ 2 \leq 2x \][/tex]
2. Divide both sides by 2:
[tex]\[ 1 \leq x \][/tex]
[tex]\[ x \geq 1 \][/tex]
### Step 2: Solve [tex]\( 2x - 8 \leq 6 \)[/tex]
1. Add 8 to both sides:
[tex]\[ 2x - 8 + 8 \leq 6 + 8 \][/tex]
[tex]\[ 2x \leq 14 \][/tex]
2. Divide both sides by 2:
[tex]\[ x \leq 7 \][/tex]
### Combine the solutions:
The solution from Step 1 is [tex]\( x \geq 1 \)[/tex] and the solution from Step 2 is [tex]\( x \leq 7 \)[/tex].
Combining these, we get:
[tex]\[ 1 \leq x \leq 7 \][/tex]
Thus, the correct answer is:
B. [tex]\( 1 \leq x \leq 7 \)[/tex]
