Solve the system of inequations: `(6x)/(4x-1)lt(1)/(2),(x)/(2x+1)ge(1)

1.	Solve the system of inequations: `(6x)/(4x-1)lt(1)/(2),(x)/(2x+1)ge(1)/(4)`.
Answer» 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.

Solve the system of inequations: `(6x)/(4x-1)lt(1)/(2),(x)/(2x+1)ge(1)/(4)`.