Question

$\left|\dfrac{x}{2} - 1\right|<3$ implies that $x$ lies in the interval

• A. $(-4, 8)$
• B. $(-3, 6)$
• C. $(-4, 6)$
• D. $(-3, 8)$

Hint:

$|x - a|<b \Leftrightarrow a - b<x<a + b$. Apply this template directly to inequalities involving absolute values.

Answer 1

The Correct Option is A

To solve the inequality \(\left|\dfrac{x}{2} - 1\right| < 3\), we need to understand the concept of absolute value inequalities. The expression \(\left| a \right| < b\) implies that the value of \(a\) lies between \(-b\) and \(b\).

Thus, the inequality \(\left|\dfrac{x}{2} - 1\right| < 3\) can be split into two inequalities:

1. \(\dfrac{x}{2} - 1 < 3\)
2. \(\dfrac{x}{2} - 1 > -3\)

We will solve these inequalities separately:

Solving the first inequality:

1. Add 1 to both sides to isolate \(\dfrac{x}{2}\): \(\dfrac{x}{2} < 4\)
2. Multiply both sides by 2 to solve for \(x\): \(x < 8\)

Solving the second inequality:

1. Add 1 to both sides to isolate \(\dfrac{x}{2}\): \(\dfrac{x}{2} > -2\)
2. Multiply both sides by 2 to solve for \(x\): \(x > -4\)

Combining these results, we find that \(-4 < x < 8\). Thus, the solution for \(x\) is the interval ( \(-4, 8\) ).

Therefore, the correct answer is \((-4, 8)\) .