Question

Which of the domain can make ∀x(x2 ≥ x) true?

Answer 1

To make the statement ∀x(x^2 ≥ x) true, the domain is (-∞, 0] ∪ [1, ∞).

Step-by-step explanation:

In order to make the statement ∀x(x^2 ≥ x) true, we need to find a domain that satisfies this condition.

Let's solve the inequality x^2 ≥ x:

1. Subtract x from both sides to get x^2 - x ≥ 0.
2. Factor the quadratic equation: x(x - 1) ≥ 0.
3. Set each factor equal to zero and solve for x to find the critical points: x = 0 and x = 1.
4. Plot these critical points on a number line.
5. Pick a test point within each interval and determine the sign of the inequality.
6. If the sign is positive, that interval is part of the solution. If the sign is negative, that interval is not part of the solution.
7. The solution is the union of the intervals where the inequality is true.

The solution is x ≤ 0 or 1 ≤ x, which means the domain that makes the statement true is (-∞, 0] ∪ [1, ∞).