To solve the inequality [tex]\( |2x - 8| < 2 \)[/tex], let's break it down step-by-step:
1. The absolute value inequality [tex]\( |2x - 8| < 2 \)[/tex] means that the expression [tex]\( 2x - 8 \)[/tex] lies within the range of -2 and 2. This can be rewritten as two separate inequalities:
[tex]\[
-2 < 2x - 8 < 2
\][/tex]
2. We now need to solve these two inequalities separately. First, let's deal with the left inequality:
[tex]\[
-2 < 2x - 8
\][/tex]
Add 8 to both sides to isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
-2 + 8 < 2x
\][/tex]
[tex]\[
6 < 2x
\][/tex]
Now, divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
3 < x
\][/tex]
So, one part of the solution is:
[tex]\[
x > 3
\][/tex]
3. Now, let's deal with the right inequality:
[tex]\[
2x - 8 < 2
\][/tex]
Again, add 8 to both sides:
[tex]\[
2x - 8 + 8 < 2 + 8
\][/tex]
[tex]\[
2x < 10
\][/tex]
Divide by 2:
[tex]\[
x < 5
\][/tex]
So, the other part of the solution is:
[tex]\[
x < 5
\][/tex]
4. Combining these two parts, we get the solution to the inequality:
[tex]\[
3 < x < 5
\][/tex]
So, the correct answer is:
A. [tex]\( 3 < x < 5 \)[/tex]
