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

34 Chapter 1 Review of algebraic techniques
Ifa>bthen
a +k >b +k
a −k >b −k
addingkto both sides
subtractingkfromboth sides
We can make similarstatements fora b,a <banda b.
When multiplying or dividing both sides of an inequality extra care must be taken.
Suppose we wish to multiply or divide an inequality by a quantityk. Ifkis positive the
inequality remains the same;ifkisnegative then the inequality isreversed.
Ifa>bthen
ka>kb
a
k > b k
⎫
⎬
⎭
ifkispositive
ka<kb
a
k < b k
⎫
⎬
⎭
ifkisnegative
Notethatwhenkisnegativetheinequalitychangesfrom >to <.Similarstatementscan
bemadefora b,a <banda b.Whenaskedtosolveaninequalityweneedtostate
all the values of the variable forwhich the inequality istrue.
Example1.34 Solve the following inequalities:
(a) 3t+1>t+7
(b) 2−3z6+z
Solution (a) 3t +1>t +7
2t +1>7
2t >6
t >3
subtractingt fromboth sides
subtracting 1 fromboth sides
dividing both sides by2
Hence all values oft greaterthan3satisfy the inequality.
(b) 2−3z 6+z
−3z 4 +z
−4z 4
z −1
subtracting 2 fromboth sides
subtractingzfrom bothsides
dividing both sides by −4,rememberingtoreverse
the inequality
Hence all values ofzgreaterthanorequal to −1 satisfy the inequality.
We often have inequalities of the form α β > 0, α < 0, αβ > 0and αβ < 0tosolve.It
β
isuseful tonote thatif
α
β >0theneitherα>0andβ>0orα<0andβ<0
α
β <0theneitherα>0andβ<0orα<0andβ>0
αβ>0theneitherα>0andβ>0orα<0andβ<0
αβ<0theneitherα>0andβ<0orα<0andβ>0
The following examples illustratethis.