If the universal set is \(U=\mathbb{R}\) and \(A=\{x\in\mathbb{R}:x^2\ge 9\}\), what is \(A'\)?
Answer and explanation
Correct answer: \((-3,3)\)
The inequality \(x^2\ge 9\) is equivalent to \(|x|\ge 3\), so \(A=(-\infty,-3]\cup[3,\infty)\). Its complement in \(\mathbb{R}\) consists of all real numbers that do not satisfy \(|x|\ge3\), namely those satisfying \(|x|<3\). Hence \(-3<x<3\), and \(A'=(-3,3)\). The endpoints are excluded because \((-3)^2=3^2=9\), so both belong to A. Therefore option A is correct.
Frequently asked questions
What is the correct answer to this question?
\((-3,3)\)
Why is this the correct answer?
The inequality \(x^2\ge 9\) is equivalent to \(|x|\ge 3\), so \(A=(-\infty,-3]\cup[3,\infty)\). Its complement in \(\mathbb{R}\) consists of all real numbers that do not satisfy \(|x|\ge3\), namely those satisfying \(|x|<3\). Hence \(-3<x<3\), and \(A'=(-3,3)\). The endpoints are excluded because \((-3)^2=3^2=9\), so both belong to A. Therefore option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
