To solve the inequality [tex]\( |x - 21| < 3 \)[/tex], we must consider the definition of absolute value.
1. The absolute value inequality [tex]\( |x - 21| < 3 \)[/tex] indicates that the distance between [tex]\( x \)[/tex] and 21 is less than 3.
2. This absolute value inequality can be split into two separate inequalities:
[tex]\[
-3 < x - 21 < 3
\][/tex]
3. We solve these inequalities separately to find the range of [tex]\( x \)[/tex]:
a. For the inequality [tex]\( -3 < x - 21 \)[/tex]:
[tex]\[
-3 < x - 21
\][/tex]
Adding 21 to both sides:
[tex]\[
-3 + 21 < x
\][/tex]
Simplifies to:
[tex]\[
18 < x
\][/tex]
b. For the inequality [tex]\( x - 21 < 3 \)[/tex]:
[tex]\[
x - 21 < 3
\][/tex]
Adding 21 to both sides:
[tex]\[
x < 3 + 21
\][/tex]
Simplifies to:
[tex]\[
x < 24
\][/tex]
4. Combining these two inequalities, we get the solution for [tex]\( x \)[/tex]:
[tex]\[
18 < x < 24
\][/tex]
Therefore, the solution to the inequality [tex]\( |x - 21| < 3 \)[/tex] is the open interval [tex]\( (18, 24) \)[/tex].
Now, on a number line, this interval is represented by two open circles at 18 and 24 (indicating that these values are not included) and a line segment connecting them, indicating all values in between are part of the solution.
Review the given options and select the one which accurately depicts the interval [tex]\( (18, 24) \)[/tex] on a number line. That will be the correct choice.
