To solve the inequality [tex]\( |x - 9| - 3 < 1 \)[/tex], let's go through it step-by-step:
1. Rewrite the Inequality:
The given inequality is [tex]\( |x - 9| - 3 < 1 \)[/tex]. First, isolate the absolute value term:
[tex]\[
|x - 9| - 3 < 1
\][/tex]
Add 3 to both sides:
[tex]\[
|x - 9| < 4
\][/tex]
2. Definition of Absolute Value:
Recall that for any real number [tex]\( a \)[/tex], [tex]\( |a| < b \)[/tex] implies:
[tex]\[
-b < a < b
\][/tex]
Apply this rule to our inequality [tex]\( |x - 9| < 4 \)[/tex]:
[tex]\[
-4 < x - 9 < 4
\][/tex]
3. Solving the Compound Inequality:
We now solve the compound inequality for [tex]\( x \)[/tex]:
[tex]\[
-4 < x - 9 < 4
\][/tex]
Add 9 to all parts of the inequality:
[tex]\[
-4 + 9 < x - 9 + 9 < 4 + 9
\][/tex]
Simplify:
[tex]\[
5 < x < 13
\][/tex]
Therefore, the solution to the inequality [tex]\( |x - 9| - 3 < 1 \)[/tex] is:
[tex]\[
5 < x < 13
\][/tex]
The correct answer is:
B. [tex]\( 5 < x < 13 \)[/tex]
