Question

Solve for x: a) `log_(0.5)(x^(2)-5x+6) ge -1`, b) `log_(1//3)(log_(4)(x^(2)-5)) gt 0`

Answer 1

a) `log_(0.5)(x^(2)-5x+6) ge 1 rArr 0 lt x^(2)-5x+6 le(0.5)^(-1)`
`rArr 0 lt x^(2) -5x+6 le2`
`{{:(x^(2)-5x+16 gt 0 rArr x^(2)-5 gt 1),(log_(4)(x^(2)-5) lt1 rArr 0 lt x^(2)-5 lt 4):} rArr 1 lt (x^(2)-5) lt 4`
`rArr6 lt x^(2) lt 9 rArr x in (-3,-sqrt(6)) cup (sqrt(6),3)`
Hence, solutions set of of original inequation: `x in (-3,-sqrt(6)) cup (sqrt(6),3)`