Question

select the correct answer.
which number line represents the solution to |x - 21| < 3?
a.
<---|---|---|---|---|---|---o---|---|---o---|---|---|---|--->
6 8 10 12 14 16 18 20 22 24 26 28 30 32
(red between 18 and 24)

b.
<---|---|---|---|---|---|---o---|---|---o---|---|---|---|--->
6 8 10 12 14 16 18 20 22 24 26 28 30 32
(red outside 18 and 24)

c.
<---|---|---|---|---|---|---•---|---|---•---|---|---|---|--->
6 8 10 12 14 16 18 20 22 24 26 28 30 32
(red between 18 and 24, filled dots)

d.
<---|---|---|---|---|---|---•---|---|---•---|---|---|---|--->
6 8 10 12 14 16 18 20 22 24 26 28 30 32
(red outside 18 and 24, filled dots)

Explanation:

Step1: Solve the inequality \(|x - 21| < 3\)

The absolute value inequality \(|a| < b\) (where \(b>0\)) is equivalent to \(-b < a < b\). So for \(|x - 21| < 3\), we have \(-3 < x - 21 < 3\).

Step2: Solve the compound inequality

Add 21 to all parts of the inequality: \(-3 + 21 < x - 21 + 21 < 3 + 21\), which simplifies to \(18 < x < 24\).

Step3: Analyze the number line representations

• Open circles are used for strict inequalities (\(<\) or \(>\)), so we eliminate options C and D (which have closed circles).
• The solution \(18 < x < 24\) means the red line should be between 18 and 24, not outside. Option A has the red line between 18 and 24 (open circles at 18 and 24), while Option B has the red line outside 18 and 24.

Answer:

A. The number line with open circles at 18 and 24 and a red line between them (representing \(18 < x < 24\))