To solve the inequality [tex]\( |x - 5| + 2 < 20 \)[/tex], we need to first isolate the absolute value expression. Here is the detailed, step-by-step solution:
1. Start by isolating the absolute value expression on one side of the inequality:
[tex]\[
|x - 5| + 2 < 20
\][/tex]
Subtract 2 from both sides:
[tex]\[
|x - 5| < 18
\][/tex]
2. Recall that the expression [tex]\( |A| < B \)[/tex] implies [tex]\( -B < A < B \)[/tex]. Apply this property to our inequality:
[tex]\[
-18 < x - 5 < 18
\][/tex]
3. Now, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex] in the middle part of this compound inequality. Add 5 to all three parts:
[tex]\[
-18 + 5 < x - 5 + 5 < 18 + 5
\][/tex]
Simplify the expressions:
[tex]\[
-13 < x < 23
\][/tex]
So, the solution to [tex]\( |x - 5| + 2 < 20 \)[/tex] is:
[tex]\[
-13 < x < 23
\][/tex]
Thus, the correct option is:
[tex]\[
-13 < x < 23
\][/tex]
