Question

9.4^-1/x +5.6^-1/x>4.9-1/x
please solve it

Answer 1

9.4^-1/x + 5.6^-1/x > 4.9^-1/x

let 2^-1/x = P , and 3^-1/x = Q

9.(2²)^-1/x + 5.(2^-1/x)(3^-1/x) -4.(3²)^-1/x

9{ 2^-1/x}² + 5(2^-1/x)(3^-1/x) -4{ 3^-1/x}² >0

9P² + 5PQ - 4Q² >0

9P² +9PQ - 4PQ - 4Q² >0

9P( P + Q) -4Q( P + Q) >0

( 9P -4Q)( P+ Q) >0

we know , exponential function always positive ,

hence,
2^-1/x >0
3^-1/x >0

hence,
( 2^-1/x + 3^-1/x) > 0 for all x belongs real numbers ,

hence,
( P + Q) > 0 always for all x€R

now,
rest
( 9P -4P) ( positive ) >0

hence,
( 9P - 4Q ) > 0

9P > 4Q

9.2^-1/x > 4.3^-1/x

9/4 > ( 3/2)^-1/x

(3/2)² > (3/2)^-1/x

3/2 is greater then 1 so, after removing no sign change ,

2 > -1/x

1/x > -2

1/x + 2 > 0

( 1 + 2x)/x > 0

so, x € ( -∞, -1/2) U ( 0 , ∞ )