082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

36 Chapter 1 Review of algebraic techniques
Itisimpossibletosatisfybotht −1andt < −2andsothiscaseyieldsnovalues
oft.
Insummary, 2t+3
t +2
1when−2<t −1.
Example1.36 Solve the following inequalities:
(a)x 2 >4 (b)x 2 <4
Solution (a) x 2 >4
x 2 −4>0
(x−2)(x+2)>0
For the product (x −2)(x +2) tobepositive requires either
(i) x−2>0andx+2>0
or
(ii) x−2<0andx+2<0.
We examine eachcase inturn.
Case(i)
Case(ii)
x−2>0andsox>2.
x+2>0andsox>−2.
Bothofthesearetrueonly whenx > 2.
x−2<0andsox<2.
x+2<0andsox<−2.
Bothofthesearetrueonly whenx < −2.
Insummary,x 2 > 4 whenx > 2 orx < −2.
(b) x 2 <4
x 2 −4<0
(x−2)(x+2)<0
For the product (x −2)(x +2) tobenegative requires either
(i) x−2>0andx+2<0
or
(ii) x−2<0andx+2>0.
We examine eachcase inturn.
Case(i)
x−2>0andsox>2.
x+2<0andsox<−2.
No values ofxarepossible.