Final answer:
To solve the inequality x³ * x ≥ 2, rearrange it as x⁴ - 2 ≥ 0. Find the critical values of x where the expression equals zero, which are ±√2. Examine the signs of x⁴ - 2 in the intervals determined by the critical values to find that x⁴ - 2 > 0 in the interval (-√2, √2). Therefore, the solution is x ∈ (-√2, √2).
Step-by-step explanation:
To solve the inequality x³ * x ≥ 2, we begin by rearranging it as x⁴ - 2 ≥ 0. Now, we need to find the critical values of x where the expression equals zero. Setting the inequality equal to zero, we get x⁴ - 2 = 0. Solving this equation, we find that x = ±√2.
Next, we need to examine the signs of the expression x⁴ - 2 in the intervals determined by the critical values. We choose test points in each interval to determine the sign. The intervals are: (-∞, -√2), (-√2, √2), and (√2, ∞). By testing these points, we find that x⁴ - 2 > 0 in the interval (-√2, √2).
Therefore, the solution to the inequality is x ∈ (-√2, √2).