Number of integers ≤10 satisfying the inequality 2log1/2⁡(x−1)≤13−1logx2−x⁡8 is

We must have x > 1 2log1/2⁡(x−1)≤13−1logx2−x⁡8 or  13−log2⁡x2−x3+2log2⁡(x−1)≥0 or  log2⁡2−log2⁡x2−x+6log2⁡(x−1)≥0 or  log2⁡2(x−1)6x(x−1)≥0 or  2(x−1)5x≥1Putting x - 1 = y, we have y > 0. Thus,  2y5y+1−1≥0 or  2y5−y−1y+1≥0 or  2y5−2y+y−1y+1≥0 or  2yy4−1+y−1y+1≥0 or  (y−1)2y(y+1)y2+1+1y+1≥0 or  y−1y+1≥0 or y≥1⇒ x≥2