What is the largest negative integer value of x, for which ∣∣​ x 3 −1 ∣∣​ ≥2−x?Enter Answer:

To solve the inequality |x^3 - 1| ≥ 2 - x, we need to consider two cases due to the absolute value:

Case 1: x^3 - 1 ≥ 2 - x Rearranging, we get x^3 + x - 3 ≥ 0 Solving this cubic inequality, we find that x ≤ -1 or x ≥ 1.73 (approximately). Since we're looking for the largest negative integer, x = -1 is the solution for this case.

Case 2: -(x^3 - 1) ≥ 2 - x Rearranging, we get x^3 + x - 1 ≤ 0 Solving this cubic inequality, we find that -1.32 ≤ x ≤ 1 (approximately). There is no negative integer solution for this case.

Therefore, the largest negative integer value of x for which |x^3 - 1| ≥ 2 - x is x = -1.