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Which number line represents the solution to [tex]|x - 21| \ \textless \ 3[/tex]?

A.
B.
C.
D.


Sagot :

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.
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