Number of integers less than or equal to 10 satisfying the inequality 2log1/2(x−1)≤13−1logx2−x8 is
We must have x>12log1/2(x−1)≤13−1logx2−x8
⇒13−log2x2−x3+2log2(x−1)≥0⇒log22−log2x2−x+6log2(x−1)≥0⇒log22(x−1)6x(x−1)≥0⇒2(x−1)5x≥1
Putting x−1=y⇒y>0⇒2y5y+1−1≥0⇒2y5−y−1y+1≥0⇒2y5−2y+y−1y+1≥0⇒2yy4−1+y−1y+1≥0⇒(y−1)2y(y+1)y2+1+1y+1≥0⇒y−1y+1≥0
⇒y≥1⇒x≥2