BMM10233 Chapter 5 Inequalities

8/11/2019 BMM10233 Chapter 5 Inequalities

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Chapter 5 | Inequalities | BMM10233 | 78 of 204

W g po d ug , , d , u quFo g qu w = qud quo

There are certain Rules of inequality

W u ddd o ud o o d o qu g o qu o g

d o

d

W v u dvdd o upd o o d o qu g o qu

Fo g d

dx y

k k

>

W -v u dvdd o upd o o d o qu g o qu g vd

Fo g d

dx y

5y

>

Inequalities


2/17


2Similarly if it is x 3x + 2 < 4x 8

(x 1 ) ( x 2 ) < 4x 8

( x 1 ) (x 2 ) < 4 (x 2 )

2

2

2

Hence we get x 1 < 4 is wrong because we do not

know the sign of x 2. So correct way is

x 3x + 2 < 4x 8

x 3x +2 4x + 8 < 0

x 7x

+ 10 < 0 and then solve for x

2

2

2 2

2 2

ax bx c 0 then

px + qx +r

ax + bx + c 0 and px + qx + r > 0

or ax + bx + c 0 and px + qx + r < 0

+ +

2

2

2 2

2 2

ax bx c 0 then

px + qx +r

ax + bx + c 0 and px + qx + r > 0

or ax + bx + c 0 and px + qx + r < 0

+ +

Inequality with algebraic expression

o w

w

ax b

0px q

+>+ d p q o d p q

Bu w gv 9 AND o vu o w qu u w o gg o o

gv qu

Howv ouo o 9 OR vu w g 9 , , up o d vu

w , , up o gv o ud

o AND 9 w w v 9 u AND intersectiond o OR 9 w w u u

OR union

Mathematical Statements

M po vovg o =, , , , , d

Inequality

A w o g o qu o o d qu o o qu , , , Io wod, po vovg o o , , , quEp: 7 v


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Identity

A op w d u o p o v o v d d

A op w d u o pu vu o v d quo

Fo p: = = 3

5x= v =

Replacement set and the solution set

Cod qu 7 T qu u, w p v u , , , , , , , -, -,

T o w vu o o d p I ov p o g

p T o vu o o p w gv d ouo

ouo = {, , , , }

Absolute value of a real number

L u o o o T po o u w opod o u

T d o po o po d ou vu o d w o-gv,

ou vu o u pov o gv w o- gv T ou vu o u w

| | ||= , | | = w o | |= , ; | |= , Tu ||= , , | |= = T ou vu o , d ou vu o o- zo u w pov

Rules for working with inequalities

Basic properties of inequalities

I ,

I > & > , h >

I > , h qui or sur o oh h sis, os o h iquli

I > , h posiiv qui mulipli or ivi o oh sis os o h iquli I > 0, h >

a bx x>

I > h giv qui mulipli or ivi o oh h sis woul rvrs h iquli I < 0, h <

a bx x , h > or + > 0 or >

I > , >

&

>

, h +

+

> +

+

I > & > , h + > +


4/17


I > & < , h >

9 I < & > , h <

0 I > , h >

Ia bc d< h < , i , > 0

1

a 2 if a > 0a+

a b

ab , if a and b are both + ve2

+

I < r o sm sig h

1 1

a b

Thus > & < 0 i < < 0h s o ll possil vlus sisig h giv iquli is : 0 < : - < < 0 = { : - < < }Thus ll h pois o h umr li lig w will sis h giv iquli


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Example: 3.

Fi h s {R / | | < }Sol:

Cs i L 0 Thror | | = I his s | | < <

< Thus < i < Cs ii wh < 0 i <

I his s | | = =

| | < < < > Thus > < i < <

Th s o ll possil vlus o sisig h giv iquli is

{R/ < < }{R/ < } = {R/ < < }

Example: 4.

Fi h s susiuig { R / < | | < }Sol:Cs i wh < | |

I 0, h | | = < | | < > I < 0, h | | = < | | < > < Thus ll vlus o < ll vlus o > will sis h iquli < | |

Thus, h s o ll rl vlus o sisig < | | is {R/ < } U {R/ >}Cs ii wh | | <

I 0, h | | = | | < < 0 < Furhr i < 0, h | | = | | < - < > < < 0

Thus {R/ < < 0}{R/0 < } h s o ll vlus o sisig h iquli < | | < is giv {R/ > }{R/ < }{R/ < < }= {R/ < < }{ R/ < < }

Example: 5.

Solv: + | | =

Sol:

Cs i wh 0, I his s || = h giv quio is + = = = Bu 0 = is lso soluio iss i his s

Cs ii l < 0I his s | | =

So, h giv quio is = or =

or =


6/17


Graphical representation on the number line

A umr li hs rl umrs mrk o i

i Th igrs r mrk o h li s show low

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

I , h > or = Soluio s = { / > } U { / = } = { / > or = } = { / }Thror, soluio s is {, , }

-4 -3 -2 -1 0 1 2 3 4

ii I < , soluio s = {/ < } = {, 0,,,,, }

iii I > <

soluio s = { / >}{ / < } = { / > < }= {/ < < }= {, 0, }

Example: 6.

Ii o h umr li, h pois , whih sis h oiio | |, whr is igr

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Sol:

Suppos, 0, h h iquli| | will ru o 0 This ms h ll h pois o h umr li ligw 0 iluig 0 , will sis his pr o h iquli

Now l < 0, h ||= I < + , h iquli is rvrs hgig h sigSi < 0, w hv < < 0

ll h pois lig w iluig h pois will sis h iquli ||< hsoluio s is {, , , , 0, , , , , }

Example: 7.

For h iquli < , whr N, h rplm s is h s o url umrs, i, NN = { , , , , }

Sol:

Th iquli < is ru, whr is rpl , , , or Thror {, , , , } is h soluio s

No: For h iquli >


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i I rplm s = { , , , , 9}, h soluio s = { , , 9}

ii I rplm s = {0, , , , }, h soluio s = {, }

Example: 8.

Lis h soluio o < giv h is posiiv igrSol:

< + <

<

<

8x 4 1x

8 8 2

> >

Th soluio s = {, , , , }N

Example: 9.

Fi h soluio s o h iquli + > 0 whr is giv igr

Sol:

-4 -3 -2 -1 0 1 2 3 4

126x 12 x x 2

6

> > >

Thror h soluio s is {-}

Example: 10.

I P is h soluio s o > + Q is h soluio s o + whr N Fi h s P QSol:

> + , > + , i, > or3

x , x 13> >

+ , i, + or + 0

20

x or x 54

x 1 and x 5> Th soluio s o P = { , , , , , , } h soluio s Q = { , , }

P Q {5, 6, 7, 8, 9, ...} N. =

Example: 11.

Solv: + < rprs h soluio grphill Z

Sol:

Th iquli + < hs wo prsi +

or 8

x, 4 x.2


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ii + <

<

< or <

, h giv iquli rus o < Thror, h soluio s = {,,,, 0, }

Th ollowig is h grphil rprsio o h soluio s

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example: 12.

Fi h soluio s o h iquli1 3

x 3x 2 , where x N4 4+ +

Sol:

3 1x 3x 2

4 4

1 52x 2 or 2 x

2 2

5x

4

1or, x 1

4

Si hr is o url umr lss h1

14

, h soluio s is mp

Example: 13.

Fi h soluio s o h iqulix 2 x 3

, where x N3 2

>

Sol:

> > 9

>9 + , > or < Thror h soluio s is {, , , }

Example: 14.

I h rplm s is { : Z} Fi h soluio s o

i >

Rplm s is {,,,,,, 0, , , , , , }

11 1

2x 11 or x i.e. x 5 . The solution set is {6}2 2

> > >

ii + + +

- or or2

x3

Thror, h soluio s is {,,,,,}


9/17


Example: 15.

I P is h soluio s o > + Q is h soluio s o + 9 , whr N, i h s P QSol:

S P : > +

> +

> , > 62

, > Thror, h soluio s is P = { , , , , }

s Q : + 9 + 9

or 184

, i, Q = {, , , }

Thror P Q = {, , , , 9 }

Example: 16.

I is giv igr, i h soluio s o2 1

(x 1) 0.5 5+ + >

Sol:

x 1 2 x 2 1or

5 5 5 5 5 5

x 3 3x 5, i.e., x 3. Therefore, the solution set is { 2, 1}

5 5 5

+ > >

> > >

Example: 17.

I W s o whol umrs, solv h iquli < 0Sol:

< 0, W = {0, , , , , , , , , }


10/17


Cs :x 4 x

2.2 3+

5x 122 or 5x 12 12, 5x 24

6

24 4 4or x x 4 or 4 x5 5 5

4Hence 4 x 6.

5

This ms h lis w4

45

Thror, h soluio s is {, }

Example: 19.

I h rplm s = {, , , , , }, lis h soluio ss o i < ii 0 < Sol:

i <

This ms, lis w is ilu, u i is o i h rplm s { , , }ii 0 < 0 < , < ,

52

< or >52

+ or 9 i, 9

2

52

< 9

2

Thror lis w52

9

2

9is included.

2

Thror, h soluio s is {, }

Example: 20.

I h rplm s = { / Z}, lis h soluio ss o i + < ii Sol:

i 2+ <

< 9, < < soluio s = {,, 0, , }

ii i, < + 9

33

9

3

Thror Th soluio s is {, , }


11/17


No: Th iquli < 0 whr < is sisi i < < h iquli > 0 is sisi

i h irvls < s wll s >

Example: 21.

I m hv rl vlu i whih is grr 3 21x 1 or x x.+ +

Sol:

( ) ( )3 2 3 2x 1 x x x x x 1 .+ + =

( )( )2x 1 x 1=

( ) ( )2x 1 x 1= +

Now ( )2

x 1 is posiiv,

3x 1+ > or 2x x+

Aorig + is posiiv or giv, h is orig s > or < I = h i quli om, quli

Example: 22.

Fi miimum vlu o(a x) (b x)

c x

+ ++

Pu c x y+ =

This will l us o

2(a c) (b c)

y a c b c 2 (a c) (b c)y

+ + +

squr rm = 0Wh (a c) (b c)=

miimum r vlu is a c b c 2 (a c) (b c) + + orrspoig vlu o x (a c) (b c) c=

Example: 23.

Fi h grs vlu o 2 3x y wh 3x 4y 5+ =

L 2 3p x y ,= lrl p, is h Prou o ors suh h wo o hm r qul o h rmiig r qul o

Now, 3x 4y 5+ =

3x 4y

2( ) 3( ) 52 9+ =

3x 3x 4y 4y 4y5.

2 2 3 3 3= + + + + =

usig wigh A M G M iquli


12/17


1

52 3

3x 4y2 3 3x 4y2 3 ( ) ( )

5 2 3

+

12 3 2 3516 3( x y ) 1 x y .

3 16

or mimum o3 3 3x y .

16=

Example: 24.

Fi soluio s o tan x 1 tan x| 3 3 | 2

Sol:

3| y | 2,

y wh tan xy 3 0= >

3 3y 2 or y 2y y

2 2y 2y 3 0 or y 2y 3 0 +

si > 0, hror , y 3 or y 1

tan x tan x3 3 or 3 1

tan x 1 or tan x 0

m z 1 1m , m

4 2

+ +

1m , m 1

2

+ +

Example: 25.

I h quio 4 3 2x 4x ax bx 1 0 + + + = hs our posiiv roos, i

Sol:

L , , our roos o h giv quio

h 4 + + + =

1 =

A M o , , ,

G.M of , , , .=

= = =

1 = = = =[ ] 4 + + + =

4 3 2 1 3 2x 4x ax bx 1 x 4x bx 4x 1 + + + = + +

a b and b 4 = =


13/17


I + < 0, h Ass Co - 0000

< < 1

2< <

4 5x

5 4<

5 4x

4 5> > No o hs

Solv or i /+ / 0 Ass Co - 0000

> Ass Co - 0000

< < / =/ 1 1

x or x 1

x2

0 > 0 No soluio

9 A Rl umr is si o lgri i i sis Polomil quio wih igrl Whih o h ollowig is olgri? Ass Co - 00009

2 3 2 0

Practice Exercise - 1


14/17


0 I x [x], x( 0), ER,= Whr [] is h grs igr lss h or qul o , h h umr o soluio o +

1f ( ) 1

x = r Ass Co - 0000

0 iii

I p, q, r r rl umr, h Ass Co - 000 m p, q = m p, q, r

mi p, q =1

2p + q |p q|

m p, q =1

2p + q |p q|

m p, q < mi p, q, r

m p, q =1

2p + q |p q|

I is rl h prssio ks ll rl vlus p hos whih li w , h r Ass Co -000

, , , , ,

I1

x 0, 0 and x 1x

> > + is lws o giv, h h ls vlu o is Ass Co - 000

1

2 0

1

4

3

4

For posiiv vlu o 2 4sin cos + lis i h irvl; Ass Co - 000

[, ] 3

, 14

1 5

,4 16

[ , ] No o hs

Th lows irvl or whih 12 9 4x x x x 1 0 + + > is Ass Co - 000

4 x 0 < 0 x 1< < 100x 100 < 0 x< < 3 x 0< <

SCORE SEET

Use HB pencil only. Abide by the time-limit

9

0


15/17


For o-giv rl umrs suh h 1 2 na a ... a p+ + + = i jq a a= , h i < j Ass Co - 000

21

q p

2

21

q p

4

> p

q

2

> 2

pq

2

> 21q p

8>

Fi h grs vlu o 3 4(a x) (a x)+ or Rl vlu o umrill lss h Ass Co - 000

3 4 7

8

6 8 a

7

2 2 7

2

6 8 a

7

3 4 7

9

6 8 a

7

3 4 8

8

6 8 a

7

3 4 7

7

6 8 a

7

I r posiiv quiis , > h Ass Co - 000

1 1

1 1 + > +

1 1

1 1 + = +

1 1

1 1 + < +

1 1

1 1 + +

No o hs

I , , r i P > h Ass Co - 0009

n n na c b+ > n n na c 2b+ > n n na c 2b+ = p, o , , n n na c 2b+

I > , whih o h ollowigs rss s irss? Ass Co - 0000

i 2x x+ ii 24x x ii1

1x

+

I ol II ol

III ol Boh I II All o h ov

I h rplm s = { , , 0, , , } lis h soluio s o 2 3x 24 x< < Ass Co - 000

{, {, , } , } , ]

2 3

4

(x 1) (x 1)0

x (x 2)

+

Ass Co - 000

1 x 2 < < 1 x 2 < 1 x 2 1 x 2 < 1 x

For posiiv rl umr , , whih o h ollowig hol Ass Co - 000

2 2 2a b c bc ca ab+ + > + +

(b c) (c a) (a b) abc+ + +

a b c 3b c a

+ +

3 3 3a b c abc+ + >

2 2 2a b c bc ca ab+ + = + +

Practice Exercise - 2


16/17


9 I 2f (x) ax bx c, 0= + + < h

I o isuss

b 1 0+ >

is giv rl is posiiv rl

b 1 0+

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BMM10233 Chapter 5 Inequalities