Q.1. Find the largest integral x which satisfies the following inequalities
1 x + 1 - 2 x 2 - x + 1 < 1 - 2 x x 3 + 1 .

Dear Student,
Please find below the solution to the asked query:

1x+1-2x2-x+1<1-2xx3+1x2-x+1-2x-2x+1x2-x+1<1-2xx+1x2-x+1x2-3x-1x+1x2-x+1<1-2xx+1x2-x+1Now for x2-x+1, coefficient of x2>0 and Discriminant<0, henceit will be greater than 0 for xR, hence it can be cancelledx2-3x-1x+1<1-2xx+1x2-3x-1x+1-1-2xx+1<0x2-3x-1-1+2xx+1<0x2-x-2x+1<0x2-2x+x-2x+1<0xx-2+1x-2x+1<0x-2x+1x+1<0Now we cancel x+1, keeping in mind that x+10 i.e. x-1x-2<0 and x-1x<2 and x-1Hence x-,2--1

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