CHAPTER 2

Equation and Inequalities

Equation

Equation are the basic mathematical tool for solving real-world problem

To solve the problem, we must know how to construct equation that model real-life situations.

Equation Linear Equation

Quadratic Equation

Polinomial equation

0bax

02 cbxax

012

21

1 ... axaxaxaxa nn

nn

Solving the Linear Equation

Solve the equation below1. Solution

8347 xx 2.

Solution

xx

4

3

3

2

6

3

124

1237

x

x

xx

7

8

4842

5448124

3

18

123

x

x

xx

xx

Modeling with Equation

Guideline for modeling with equation

Identify the variable Express all unknown quantities in

term of the variable Set up the model Solve the equation and check your

answer

Example 1A square garden has a walkway 3m wide around its outeredge. If the area of the entire garden, including thewalkway, is 18,000m2, what are thedimensions of planted area?

Solution:We are asked to find the length and width of the planted area. So we let x = the length of the planted area

Next, translate the information into the language of algebra

We now set up the model. area of entire garden = 18, 000m2

The planted area of the garden is about 128m by 128m

In word In Algebra

Length of planted area xLength of entire garden x + 6Area of entire garden (x + 6)2

128

6000,18

000,186

000,186 2

x

x

x

x

Example 2A manufacturer of soft drinks advertise their orange soda as ‘natural flavored’although it contains only 5% orange juice.A new federal regulation stipulated that to be called ‘natural’a drink must contain at least 10% fruit juice. How muchpure orange juice must this manufacturer add to 900 gal oforange soda to conform to the new regulation?

SolutionThe problem asks for the amount of pure orange juice to beadded. So letx = the amount (in gallons) of pure orange juice to be added

Next, translate the information into the language of algebra

To set up the model, we use the fact that the total amount oforange juice in the mixture is equal to the orange juice in the first two vats. Amount of amount of amount of Orange juice + orange juice = orange juice In first vat in second vat in mixture

The manufacturer should add 50 gal of pure orange juice to the soda

In word In Algebra

Amount of orange juice to be added xAmount of the mixture 900 + xAmount of orange juice in the first vat (0.05)(900) = 45Amount of orange juice in the second vat (1)(x) = xAmount of orange juice in the mixture 0.10(900 + x)

509.0

45

459.0

1.09045

)900(1.045

x

x

xx

xx

Solving Quadratic Equations By factoring1. Solve the equation Solution:

By completing the square2. Solve the equation Solution:

2452 xx

8@3

083

02452

xx

xx

xx

06123 2 xx

)43(6443

643

6123

2

2

2

xx

xx

xx

22

22

22

6232

2

x

x

x

x

Modeling with Quadratic EquationsA farmer has rectangular garden plot surrounded by 200mof fence. Find the length and width of the garden if itsarea is 2400m2.Solution:We are asked to find the length and width of the garden. Solet w = width of the garden

Now set up the model.(width of garden).(length of garden) = area of garden

The dimension of the garden is 60m by 40m

In word In Algebra

Width of garden wLength of garden (200–2w)/2 = 100-w

40@60

04060

02400100

24001002

ww

ww

ww

ww

Polynomial Equations Polynomial equation can be solve by change it into

quadratic equation. Solve by substituting If then

When Then so or

When Then so

0122 234 xxxxx

xu1

.

xxu1

222 1

2x

xu

0122 234 xxxx :2x 0

1212

22

xxxx

1@3

013

032

0122

011

21

2

2

22

uu

uu

uu

uu

xx

xx

3u 31

x

x2

53x

1u 11

x

x2

31 ix

Application Energy Expended in Bird Flight

Ornithologist have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours, because air generally rises over land and falls over water in the daytime, so flying over water requires more energy. A bird is released from point A on an island, 5 mi from B, the nearest point on a straight shoreline. The bird flies to a point C on the shoreline and then flies along the shoreline to its nesting area D. Suppose the bird has 170 kcal of energy reserves. It uses 10 kcal/mi flying over land and 14 kcal/mi flying over water .(a) Where should the point C be located so that the bird use exactly 170kcal of energy during its flight?

(b) Does the bird have enough energy reserves to fly directly from A to D?

Solution:(a) We are asked to find the location of C. So let

x = distance from B to CFrom the fact that

energy used = energy per mile X miles flownWe determine the following:

Now we set up the model.total energy used = energy used over water + energy used over land

To solve this equation, we eliminate the square root by first bringingall other terms to the left of the equal sign and then squaring eachside

In word In Algebra

Distance from B to C xDistance flown over water (from A to C) Distance flown over land (from C to D) 12 – xEnergy used over water 14Energy used over land 10(12-x)

xx 12102514170 2

252 x

252 x

Point C should be either 6(2/3)mi or 3(3/4)mi from B so that the bird uses exactly 170 kcal of energy during its flight.

(b) By the Pythagorean Theorem, the length of the route directly from A to D is √52+122 = 13mi, so the energy the bird requires for that route is 14 x 13 = 182 kcal. This is more energy than the bird has available, so it can’t use this route.

4

33@

3

26

02400100096

490019610010002500

25141050

25141050

25141210170

2

22

222

2

2

xx

xx

xxx

xx

xx

xx

Simultaneous Equation Simultaneous equation has two or more equation that has

similar set of solution. Linear equation can be solved by using substitution and elimination method.

Modeling with Linear Systems

Guideline for modeling with systems of equations1. Identify the variable.2. Express all unknown quantities in

terms of variables.3. Set up a system of equations.4. solve the system and interpret the results.

ExampleA researcher performs an experiment to test the hypothesis thatinvolves the nutrients niacin and retinol. She feeds one group oflaboratory rats a daily diet of precisely 32 units of niacin and22000 units of retinol. She uses two types of commercial palletfoods. Food A contains 0.12 unit of niacin and 100 units of retinolper gram. Food B contains 0.20 unit of niacin and 50 units ofretinol per gram. How many grams of each food does she feed thisgroup of rats each day?

Solution:

For food A 200 grams and for food B 40 grams

200

40

56014

)4(2640612

)3(32002012

)2(000,2250100

)1(3220.012.0

x

y

y

yx

yx

yx

yx

ExampleA farmer has 1200 acres of land on which he grows corn,banana and watermelon. It costs RM45 per acre to growwatermelon, RM60 for corn and RM50 for banana. WithRM63 750 how many acres of each crop can be planted ifthe acreage of corn planting twice as watermelon?

Solution:)1(1200 zyx

)2(63750506045 zyx)3(2 yx

)4(12003 zx)5(6375050165 zx

)6(6000050150 zx375015 x250x

500)250(2 y450)500250(1200 z

Partial Fraction

The denominators of the algebraic fractions encountered

will be of three basic types:

Inequalities

Linear inequality with one variable

Two linear inequality with one variable

Rational inequality

Quadratic inequality

Modeling with InequalitiesExample Students in Animal Husbandry Science from Faculty of AgroIndustry and Natural Resources, Universiti MalaysiaKelantan planning to held an Animal Carnival. One of theactivities in the Carnival is riding the horse. Thestudents who handle this activity has two plans for tickets

Plan A:RM5 entrance fee and 25sen each ridePlan B: RM2 entrance fee and 50sen each ride

How many rides would you have to take for plan A to beless expensive than plan B?

Solution: cost plan A < cost plan B x = number of ridesCost with plan A = 5 + 0.25xCost with plan B = 2 + 0.50x

So if you plan to take more than 12 rides, plan A is lessexpensive

12

325.0

5.0225.05

x

x

xx

Example

A ticket for a Biotechnology exhibition is RM50 per person. Areduction of 10 cent per ticket will be given to students whocome by group. A group of UMK students decides to attendthe exhibition. The cost of chartering the bus is RM450,which is to be shared equally among the students. Howmany students must be in the group for the total cost (busfare and exhibition ticket) per student to be less than RM54?

Solution: We were asked for the number of students in that group. Solet x = number of students in the group. The information inthe problem maybe organized as follows:

In words In algebra

Numbers of students in a group x

Bus cost per student 450/x

Ticket cost per student 50 – 0.1x

Now we set up the model;Bus cost per student + Ticket cost per student < 54

-90 0 50

Set of solution:There is no negative number of student so that the groupmust have more than 50 students so that the total costRM54

0

5090

0404500

54)10.050(450

2

x

xxx

xx

xx

x

xx )50(90

(90 + x) - + + +

(50 - x) + + + -

x - - + +

+ - + -

),50(0,90