Solve the inequation `2^(2x^(2)-10x+3)+6.2^(x^(2)-5x).3^(x^(2)-5x)ge3^(2x^(2)-10x+3)`

Explanation

Thegiven inequation is equivalent to
`8.2^(2(x^(2)-5x))+6.2^(x^(2)-5x).3^(x^(2)-5x)-27.3^(2(x^(2)-5x))ge0`
Let `2^(x^(2)-5x)=f(x)` and `3^(x^(2)-5x)=g(x)`
then `8.f^(2)+6f(x).g(x)-27g^(2)(x)ge0`
On dividing in each by `g^(2)(x) [ :' g(x)gt0]`
Then `8((f(x))/(g(x)))^(2)+5((f(x))/(g(x)))-27ge0`
and let `(f(x))/(g(x))=t[:'tgt0]`
then `8t^(2)+6t-27ge0`
`implies(t-3/2)(t+9//4)ge0`
`impliestge3//2` and `tle-9//4`
The second inequation has no root. `[:'tge0]`
From the first inequation `tgt3//2`
`(2/3)^(x^(2)-5x)ge(2/3)^(-1)[ :' 2/3 lt1]`
`impliesx^(2)-5xle-1impliesx^(2)-5x+1le0`
`:.(5-sqrt(21))/2lexle(5+sqrt(21))/2`
Hence `x epsilon [(5-sqrt(21))/2,(5+sqrt(21))/2]`