Question

Solve the inequation `log_((x-3))(2(x^(2)-10x+24)gelolg_((x-3))(x^(2)-9)`

Answer 1

This inequation is equivalent to the collection of systems
`{(2(x^(2)-10x+24)gex^(2)-9),(x^(2)-9gt0),(x-3gt1):}`
`{(2(x^(2)-10x+24)lex^(2)-9),(2(x^(2)-10x+24)gt0),(0ltx-3lt1):}`
On solving the first system, we have
`{(x^(2)-20x+57ge0),((x+3)(x-3)gt0),(xgt4):}`
`hArr {(x epsilon(-oo,10-sqrt(43)]uu[10+sqrt(43),oo)),(x epsilon (-oo,-3)uu(3,oo)),(x epsilon(4,oo)):}`
Therefore, the system has solution
`xge10+sqrt(43)`
i.e. `x epsilon[10+sqrt(43),oo)`