Let's analyze the given sets \(A\) and \(B\) and determine the correct option.
Set A: The set is defined as \(A = \{x : |x^2 - 10| \le 6\}\).
- This implies \(-6 \le x^2 - 10 \le 6\).
- Solving the inequalities:
- For \(x^2 - 10 \ge -6\), we get \(x^2 \ge 4\). Thus, \(x \le -2\) or \(x \ge 2\).
- For \(x^2 - 10 \le 6\), we get \(x^2 \le 16\). Thus, \(-4 \le x \le 4\).
- Combining these, we have \(A = [-4, -2] \cup [2, 4]\).
Set B: The set is defined as \(B = \{x : |x - 2| > 1\}\).
- This implies \(x - 2 > 1\) or \(x - 2 < -1\).
- Solving the inequalities:
- \(x - 2 > 1\) implies \(x > 3\).
- \(x - 2 < -1\) implies \(x < 1\).
- Thus, \(B = (-\infty, 1) \cup (3, \infty)\).
We need to determine \(A \cup B\).
From \(A = [-4, -2] \cup [2, 4]\) and \(B = (-\infty, 1) \cup (3, \infty)\), we can find the union:
- \([-4, -2]\) is within \((-\infty, 1)\).
- \([2, 4]\) overlaps with \((3, \infty)\).
- The union gives us \((-\infty, 1] \cup (2, \infty)\).
Thus, the correct answer is \(A \cup B = (-\infty, 1] \cup (2, \infty)\).
Similar Questions
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
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Important Questions from Algebra
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
