video math tutor: basic math: lesson 2 - equalities & inequalities

8/14/2019 Video Math Tutor: Basic Math: Lesson 2 - Equalities & Inequalities

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B A S I C M A T H A S elf-T utorial

by

L uis A nthony A st P rofessional M athematicsT utor

L ESSON 2 :

E QUALITIES & I NEQUALITIES

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ORDER OF N UMBERS Num bers ar e order ed relative to where t hey ar e in relat ion t o zero.Num bers to the right of zero (on t he n um ber line) ar e called Pos i t ivenum bers and th ose to the left a re Nega t ive numbers .

|

Negative 0 Positive

( To type in negat ive nu mbers, use t he n egationkey and the number keys : , , , , e tc. Use t he key for ze ro, butyou should NOT type th e key in front of a num ber when you wan t toinput a posit ive num ber into the calculator. Just type the nu mber. Forexam ple, To type in th e expression: +3 4 you sh ould pr ess t he followingkeys: . . Dont t ype: . You willget all sort of str an ge results with t his.

Of cour se, th is does n ot ap ply to addition of nu mber s. To type 1 + 2 youtype: .

E QUA = LITIES A nu mber is E q u a l to another if it is locat ed at th e same point as a nothernum ber on th e real number l ine. For exam ple, the nu mber a is equa l tothe number b if th ey can be gra phed on th e sam e point on a r eal nu mberline:

a H

bThe above represen ts:

( On simple, non-gra phin g calculat ors , th e key isused t o have th e calculat or figur e out wha t is equal to th e expression youtyped into it. For graph ing calculat ors, use th e key. There is anequa ls symbol = th at is available on graph ing calculat ors, but it is used


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for compa risons bet ween t wo sides of an expression. Th is key can be foun din th e TEST Menu th at is accessed by pressing y < TEST =

P R O B L E M 1: Is 0 .375 equa l to ?

Wh a t t o Do: On t h e Ca lcu l a t or S cr een :T h e E a s y Wa y : J ust type in th e

an d see if it equa ls to 0 .375:

Yep! It s t he s ame!

T h e M o r e C o m p l ic a t e d Wa y :Compa re 0 .375 to using th e

= in t he TE ST Menu :

y < TEST =

FThe more

com plicat ed way is u sed m ostlywhen creat ing program s, sodont worr y about it t oo much.

The result of mean s you t yped in aTRUE sta tement, so the nu mbers areequa l. If the result ha d been , thenit would ha ve been FALSE, and t henu mbers would ha ve been not euqual.

O 0

P R O B L E M 2: Is equal to ?

Wh a t t o Do: On t h e Ca lcu l a t or S cr een :

No, they ar e not equal.


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I NEQUA ITIES F A nu mber is Les s T h an an other if it s point on the n um ber line is to th eleft of th e oth er.

M a t h S y m b o l fo r i s le s s t h a n i s :

For exam ple, nu mber a is less tha n a n um ber b ( ) if t he poin trepresenting a can be graph ed to th e left of point wh ere b is on t he r ealnum ber l ine:

H H

a bThe above represen ts:

F A nu mber is G r ea t e r T h a n an oth er if it is to th e right of the oth ernumber.

M a t h S y m b o l fo r is g r e a t e r t h a n i s:

For exam ple, nu mber a is great er th an a n um ber b ( ) if th e pointrepresenting a can be graph ed to the r ight of point wh ere b is on t he rea lnum ber l ine:

H

H

b aThe above repr esents:

HOT TIP!

It s ea sy to mix up th e symbols, so just rem ember th atth e openin g of th e symbol is always t owar ds t he la rgervalue. The point is toward s th e sma ller va lue. Anoth erway to remember is th at you can u se the t o wr iteth e words ess t han. The less t ha n symbol looks likethe letter L . You cant do t his wit h t he othe r symbol.

( You can access t he inequa lity k eys by accessingth e TEST Menu . Press : y < TEST = to see t he following screen aga in:


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Try u sing t his screen for th e following problem:

P R O B L E M 3: Is ?

Wh a t t o Do: On t h e Ca lcu l a t or S cr een :

y < TEST =

Since th e result is zero, this mea ns weget a false sta tem ent; th erefore, is

NOT greater tha n .

F To solve th e pr evious pr oblem by ha nd, you

would need t o conver t t he fractions t o decima ls, th en comp ar e th em.

Man y symbols ar e used to compa re one quan titywith an oth er. Here is a ta ble of th e most comm onones, along with how to rea d th e symbols, an dhow they a re u sed in possible word problems:

H ow t o sa y: H ow u sed in w or d p r ob lem s: E x a m p le:

is equ a l to is, is equ a l to or was 2 + 2 = 4

is not equ al t oNot usu ally in word pr oblems.

Used mostly t o check va lidity of problem or t o make a rest riction.

Given: , th en x 0

is

approximatelyequal t o

Used to cha nge exact an swer to anapproximat e answer.

is less t ha n is less th an or fewer th an 3 5 is grea ter th an is gr ea ter th an or more th an 2 7

is less th an orequal t o

no more t ha n or at m ostYou can work at most

40 hr s/wk. W 40

is great er th anor equal t o

no less th an or at least You mu st be at least

21 to enter h ere. A 21

Now let u s put all this together in th e next section.


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[I NTERVAL ], I NEQUA ITY &

{S ET N OTATION } Often , th e results of ma th problems are a spa n of nu mbers on the nu mberline. We call th ese ar eas I n t e rva l s . There ar e th ree principal ways of representing these values tha t a ppear on the graph of a r eal number l ine:

I n t e rva l No ta t i on , I n e q u a l i t y N ot a t i o n and S e t N o t a t i on .

THE BUILDING BLOCKS:If th e endpoint includes t he valu e, th en a solid circle or dot is u sed: H

If th e endpoint does N OT include th e valu e, th en an open circle oropen dot is u sed:

The n um ber line is sha ded (th icker line: 66 ) wher e values a re included.

A regular line ( ) is used to show th e par t of th e nu mber line th at is not

part of what we wan t.

When wha t we wan t extends to eith er t he left indefinitely, or t o th e rightindefinit ely, we use t he in finity symbol ( ) to represent th is. We usefor the left, a nd just (which is positive) for the right side of th e nu mberline.

Her es a ha ndy litt le table th at helps yourem ember h ow all th e symbols are r elated:

I f you see< or >

I f you see< or

The dot on nu mber line is open

The dot on n umber line is closed H

Always use Par ent heses:( or )

Always use Squa re Br ackets:[ or ]

When u sing th e infinity symbols, ALWAYS use parenth eses:( or )


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Her e is a t able of most of th e basic int erva ls youwill encoun ter :

N u m b er L i n eG r a p h

In te rva l No ta t ion

I n e q u a l i t y Nota t ion

S e t N o t a t i o n H o w t o R e a d t h e

S e t N o t a t i o n

566 a

( , a ) x a { x | x a }

The set of all xs

such th at x is lessthan a

566H b

( , b ] x < b { x | x < b}

The set of all xssuch th at x is lesstha n or equal to b

664

c

(c, ) x c { x | x c}The set of all xssuch th at x isgreater tha n c

H664

d

[d , ) x d { x | x d }

The set of all xssuch th at x isgreater tha n or

equal to d 666

a b(a , b ) a x b { x | a x b}

The set of all xssuch th at x is inbetween a an d b

H666 c d

[c, d )c < x d { x | c < x d }

The set of all xssuch th at x is inbetween c an d d ,an d includes c

666H e f

(e, f ]e x < f { x | e x < f }

The set of all xssuch th at x is inbetween e an d f ,

an d includes f

H666H g h

[g , h ] g < x < h { x | g < x < h }

The set of all xssuch th at x is inbetween g an d h ,inclusive

5666664 ( , ) NotApplicable

{ x | x R }

The set of all xssuch th at x is anelemen t of the set

of real nu mbers>

.. There ar e other num ber l ine graphs tha t a re possible, but thesea re covered in th e Algebra Lesson: Solving Linea r Inequ alities.


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A DET AILED EXAMP LE: Sa y we are given th e following nu mber linegraph:

5666

2

I will use x to represent the sha ded part on t he nu mber l ine graph above.

Now, lets figure out t he int erva l, inequa lity an d set n ota tion associat edwith th e given graph .

When creat ing the i n t e r v a l n o t a t i o n , always view it from t he per spectiveleft-to-right . Fr om t he gr ap h a bove, th e inter val nota tion describing it is:( , 2). This is classified as a n open int erva l, since pa ren th eses ar e usedat both ends of th e interval.

For t he i n e q u a l it y n o t a t i on , note th at all x values are less th an (but n otequa l to) 2, since an open dot is u sed. The nota tion is: x 2.

Th e s e t n o t a t i on would look like: { x | x 2} This is r ea d a s: The set of all x values such t ha t x is less th an two.

ANOTHER EXAMP LE: We ar e now given t he following nu mber linegraph:

H6666

3 1

I will use x again to represent th e shaded part on t he num ber line gra phabove. Now, lets figur e out th e inter val, inequ ality a nd set nota tionassociated with th e given graph .

Th e i n t e r v a l n o t a t i on describin g it is: [3, 1). This is classified as a h a lf-open int erval, since a pa ren th esis is used only at one end of the inter val.

For t he i n e q u a l it y n o t a t i on , note th at all x values are between 3 and 1,but dont in clude 1. To writ e th e inequa lity nota tion, place th e x between

th e two num bers. Use inequality symbols between t he x

and the num bers .The notation looks like: 3 < x 1.

Th e s e t n o t a t i on is: { x | 3 < x 1} Th is is r ea d a s: The set of all xvalues such t hat x is between n egative thr ee an d one, an d includesnegat ive thr ee.


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L ESSON 2 Q UIZ 1 Using x to represent the sh aded par t on th e num ber l ine graph below,write th e inter val, inequality and set n ota tion a ssociated with th e givengraph:

66644

Interval Notation: ___________

Inequality Notation: ___________

Set Notation: __________________

2 Using x to represent the sh aded part on t he n umber l ine gra ph below,write th e inter val, inequality and set n ota tion a ssociated with th e givengraph:

H66666H

0.3 4 .7Interval Notation: ___________

Inequality Notation: ___________

Set Notation: __________________

3 Fill in t he following t able: N u m b er L i n e

G r a p h I n t e r v a l N o t a t i o n



5666 6

[1, )

x <

{ x | 0.6 < x 5.9}

A NSWERS O N N EXT P AGE


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A NSWERS 1 Using x to represent the sh aded par t on th e num ber l ine graph below,write th e inter val, inequality and set n ota tion a ssociated with th e givengraph:

6664

4Int erval Nota tion: (4, ` )

Inequa lity Notat ion: x . 4

Set Notat ion: { x | x . 4}

2 Using x to represent the sh aded part on t he n umber l ine gra ph below,write th e inter val, inequality and set n ota tion a ssociated with th e given

graph: H66666H

0.3 4 .7Int erval Nota tion: [0 .3, 4 .7]

Inequa lity Notat ion: 0 .3 < x < 4 .7

Set Notat ion: { x | 0 .3 < x < 4.7}

3 H e r e i s t h e f il le d t a b l e : N u m b er L i n e

G r a p h I n t e r v a l N o t a t i o n



5666 6

( , 6 ) x 6 { x | x 6}

H6664

1[1, ) x 1 { x | x 1}

66666H

x < { x |

x < }

H66666

0.6 5 .9[0 .6, 5 .9) 0 .6 < x 5.9 { x | 0.6 < x 5.9}

E ND O F L ESSON 2

video math tutor: basic math: lesson 2 - equalities & inequalities

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