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CHAPTER 2
Equation and Inequalities
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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.
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Equation Linear Equation
Quadratic Equation
Polinomial equation
0bax
02 cbxax
012
21
1 ... axaxaxaxa nn
nn
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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?
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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
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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
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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.
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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.
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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
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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
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Partial Fraction
The denominators of the algebraic fractions encountered
will be of three basic types:
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Inequalities
Linear inequality with one variable
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Two linear inequality with one variable
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Rational inequality
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Quadratic inequality
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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
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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
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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