Operations On Sets

Given,

x%5E2-%7Cx%2B2%7C%2Bx%3E0%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20...(i)

Case I When

%5Chspace%7B20mm%7Dx%2B2%5Cge0

%5Ctherefore%5Chspace%7B20mm%7Dx%5E2-x-2%2Bx%3E0%20%5CRightarrow%20x%5E2-2%3E0

%5CRightarrow%5Chspace%7B25mm%7Dx%5Csqrt2

%5CRightarrow%5Chspace%7B25mm%7Dx%5Cin(-2%2C-%5Csqrt2)%5Ccup(%5Csqrt2%2C8)%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20...(ii)

Case II When

x%2B2%3C0

%5Ctherefore%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20x%5E2%2Bx%2B2%2Bx%3E0

%5CRightarrow%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20x%5E2%2B2x%2B2%3E0

%5CRightarrow%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20(x%2B1)%5E2%2B1%3E0

which is true for all x.

%5Ctherefore%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20x%5Cle-2%20%5C%2C%20%5C%2C%20or%5C%2C%20x%5Cin(-8%2C-2)%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20%5C%2C%20...(iii)

From Eqs. (ii) and (iii), we get

%5Chspace%7B25mm%7Dx%5Cin(-8%2C-%5Csqrt2)%5Ccup(%5Csqrt2%2C8)

Chapter Questions
1 MCQs