To solve the inequality [tex]\( |2x - 4| \leq 8 \)[/tex], we need to consider the definition of absolute value, which states that [tex]\( |a| \leq b \)[/tex] if and only if [tex]\( -b \leq a \leq b \)[/tex].
Given [tex]\( |2x - 4| \leq 8 \)[/tex], we can write this as:
[tex]\[
-8 \leq 2x - 4 \leq 8
\][/tex]
We will solve this compound inequality in two steps.
### Step 1: Solving the left inequality
[tex]\[
-8 \leq 2x - 4
\][/tex]
Add 4 to both sides:
[tex]\[
-8 + 4 \leq 2x - 4 + 4
\][/tex]
[tex]\[
-4 \leq 2x
\][/tex]
Divide both sides by 2:
[tex]\[
-\frac{4}{2} \leq \frac{2x}{2}
\][/tex]
[tex]\[
-2 \leq x
\][/tex]
### Step 2: Solving the right inequality
[tex]\[
2x - 4 \leq 8
\][/tex]
Add 4 to both sides:
[tex]\[
2x - 4 + 4 \leq 8 + 4
\][/tex]
[tex]\[
2x \leq 12
\][/tex]
Divide both sides by 2:
[tex]\[
\frac{2x}{2} \leq \frac{12}{2}
\][/tex]
[tex]\[
x \leq 6
\][/tex]
### Combining the results
Combining the two parts of the compound inequality, we get:
[tex]\[
-2 \leq x \leq 6
\][/tex]
Therefore, the solution to [tex]\( |2x - 4| \leq 8 \)[/tex] is:
[tex]\[
x \geq -2 \text{ and } x \leq 6
\][/tex]
This matches option B.
So, the answer is:
B. [tex]\( x \geq -2 \text{ and } x \leq 6 \)[/tex]
