Higher Engineering Mathematics

INEQUALITIES 15

A

It is not possible to satisfy bothx≥−1 and

x<−2 thus no values ofxsatisfies (ii).

Summarizing,

2 x+ 3

x+ 2

≤1 when− 2 <x≤− 1

Now try the following exercise.

Exercise 10 Further problems on inequali-

ties involving quotients

Solve the following inequalities:

1.

x+ 4

6 − 2 x

≥0[− 4 ≤x<3]

2.

2 t+ 4

t− 5

>1[t>5ort<−9]

3.

3 z− 4

z+ 5

≤2[− 5 <z≤14]

4.

2 −x

x+ 3

≥4[− 3 <x≤−2]

2.5 Inequalities involving square

functions

The following two general rules apply when inequal-

ities involve square functions:

(i)ifx^2 >k thenx>

√

kor x<−

√

k (1)

(ii)ifx^2 <k then−

√

k<x<

√

k (2)

These rules are demonstrated in the following

worked problems.

Problem 9. Solve the inequality:t^2 > 9

Sincet^2 >9 thent^2 − 9 >0, i.e. (t+3)(t−3)>0by

factorizing

For (t+3)(t−3) to be positive,

either (i) (t+3)> 0 and (t−3)> 0

or (ii) (t+3)< 0 and (t−3)< 0

(i) If (t+3) > 0 thent>−3 and if (t−3)>0 then

t> 3

Both of these are true only whent> 3

(ii) If (t+3)<0 thent<−3 and if (t−3)<0 then
t< 3
Both of these are true only whent<− 3

Summarizing,t^2 >9 whent>3ort<− 3

This demonstrates the general rule:

ifx^2 >k then x>

√

k or x<−

√

k (1)

Problem 10. Solve the inequality:x^2 > 4

From the general rule stated above in equation (1):

ifx^2 >4 thenx>

√

4orx<−

√

4

i.e. the inequality:x^2 >4 is satisfied whenx>2or

x<− 2

Problem 11. Solve the inequality:

(2z+1)^2 > 9

From equation (1), if( 2 z+ 1 )^2 >9 then

2z+1>

√

9or2z+ 1 <−

√

9

i.e. 2 z+ 1 >3or2z+ 1 <− 3

i.e. 2 z>2or2z<−4,

i.e. z> 1 or z<− 2

Problem 12. Solve the inequality:t^2 < 9

Sincet^2 <9 thent^2 − 9 <0, i.e. (t+3)(t−3)<0by

factorizing. For (t+3)(t−3) to be negative,

either (i) (t+3)> 0 and (t−3)< 0

or (ii) (t+3)< 0 and (t−3)> 0

(i) If (t+3)>0 thent>−3 and if (t−3)<0 then

t< 3

Hence (i) is satisfied when t>−3 andt< 3

which may be written as:− 3 <t< 3

(ii) If (t+3)<0 thent<−3 and if (t−3)>0 then

t> 3

It is not possible to satisfy botht<−3 andt>3,

thus no values oftsatisfies (ii).

Summarizing,t^2 <9 when− 3 <t< 3 which means

that all values oftbetween−3 and+3 will satisfy

the inequality.

This demonstrates the general rule:

ifx^2 <kthen−

√

k<x<

√

k (2)

Problem 13. Solve the inequality:x^2 < 4