Final answer:
To find the set of all real numbers satisfying the given inequality and the condition |x| ≤ 1, we transform the inequality into a quadratic equation and solve using the quadratic formula. The roots are then checked against the condition to determine the valid solutions.
Step-by-step explanation:
We have the inequality √(1 - x) * 2 ≥ x * 2 where |x| ≤ 1.
First, we need to square both sides to remove the square root, giving us (1 - x) * 2 ≥ (x * 2)². This simplifies to 2(1 - x) ≥ 4x². We then rewrite this inequality as a quadratic equation: 0 ≥ 4x² + 2x - 2.
Using the quadratic formula x = [-b ± √(b² - 4ac)] / (2a), we find the roots of the equation. After simplification, we get two solutions for x.
However, since |x| ≤ 1, we must check that these solutions are within the given range. After evaluating both roots, we determine which solutions are valid given the condition |x| ≤ 1 and the original inequality.