Question

select the correct answer.
which number line represents the solution to |x - 5| < 3?
a.
<--(-10)-(-8)-(-6)-(-4)-(-2)-0-2(open circle)-4-6-8(open circle)-10-12-14-16-->

b.
<--(-10)-(-8)-(-6)-(-4)-(-2)-0-2(open circle)-4-6-8(open circle)-10-12-14-16-->
(red line on left of 2 and right of 8)

c.
<--(-10)-(-8)-(-6)-(-4)-(-2)-0-2(closed circle)-4-6-8(closed circle)-10-12-14-16-->
(red line between 2 and 8)

d.
<--(-10)-(-8)-(-6)-(-4)-(-2)-0-2(closed circle)-4-6-8(closed circle)-10-12-14-16-->
(red line on left of 2 and right of 8)

Explanation:

Step1: Solve the absolute - value inequality

The absolute - value inequality is \(|x - 5|\lt3\). By the definition of absolute value, if \(|a|\lt b\) (where \(b\gt0\)), then \(-b\lt a\lt b\). So for \(|x - 5|\lt3\), we have \(- 3\lt x - 5\lt3\).

Step2: Solve the compound inequality

To solve \(-3\lt x - 5\lt3\), we add 5 to all parts of the inequality.
For the left - hand side: \(-3 + 5\lt x-5 + 5\), which simplifies to \(2\lt x\).
For the right - hand side: \(x-5 + 5\lt3 + 5\), which simplifies to \(x\lt8\).
So the solution to the inequality \(|x - 5|\lt3\) is \(2\lt x\lt8\).

Step3: Analyze the number - line representations

• For an inequality of the form \(a\lt x\lt b\) (where \(a = 2\) and \(b = 8\) in our case), the number - line should have open circles at \(x = 2\) and \(x = 8\) (because the inequality is strict, \(x

eq2\) and \(x
eq8\)) and the line should be shaded between \(2\) and \(8\).

• Option A: Has open circles at \(2\) and \(8\) and is shaded between \(2\) and \(8\), which matches our solution.
• Option B: Is shaded outside the interval \(2\) to \(8\), which is incorrect.
• Option C: Has closed circles at \(2\) and \(8\), but our inequality is strict (\(x

eq2\) and \(x
eq8\)), so closed circles are wrong.

• Option D: Is shaded outside the interval \(2\) to \(8\) and has closed circles, which is incorrect.

Answer:

A. The number line with open circles at 2 and 8 and shaded between 2 and 8.