The first inequation of the given system is
`(6x)/(4x-1)lt(1)/(2)`
`(6x)/(4x-1)-(1)/(2)lt0`
`rArr (12x-4x+1)/(2(4x-1))lt0`
`rArr (8x+1)/(2(4x-1))lt0rArr(8x+1)/(4x-1)lt0." "...(i)`
`therefore (8x+1gt0and 4x-1lt0) or (8x+1lt0and 4x-1gt0)`
`rArr(8xgt-1and 4xlt1) or(8xlt-1 and 4xgt1)`
`rArr (xgt(-1)/(8)and xlt(1)/(4))or(xlt(-1)/(8)and xgt(1)/(4))`
`rArr (-1)/(8)ltxlt(1)/(4)" "[therefore xlt(-1)/(8)and x gt(1)/(4)" is not possible"]`
`rArr x in((-1)/(8),(1)/(4))`.
The equation inequation of the given system is
`(x)/(2x+1)ge(1)/(4)`
`rArr(x)/(2x+1)-(1)/(4)ge0`
`rArr (4x-2x-1)/(4(2x+1))ge0rArr(2x+1)/(2x+1)ge0" "...(ii)`
`therefore (2x-1le0and 2x+1lt0) or(2x-1ge0and2x+1gt0)`
`rArr (xle(1)/(2)and xlt(-1)/(2))or(xge(1)/(2)and x gt(-1)/(2))`
`rArr (xlt(-1)/(2))or(xge(1)/(2))`
`rArr x in(-oo,(-1)/(2))orx in[(1)/(2),oo)`
`rArr x in(-oo,(-1)/(2))uu[(1)/(2),oo)`
`therefore " solution set " =((-1)/(8),(1)/(4))nn{(-oo,(-1)/(2))uu[(1)/(2),oo)}=phi`.
Hence, the given system of inequation has no solution.
