Math 1 Module 2

8/3/2019 Math 1 Module 2

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Module 2.5 : First-Degree Equations and Inequalitiesin One Variable

EXPLORE Your Understanding

Activity 1

Do You Remember?

This module is about first-degree equations and inequalities inone variable. As you go through the activities/exercises, you willbe able to identify and translate mathematical sentences into first-

degree equations and inequalities in one variable, and describe situations where equations and inequalities are

used.

ALGEBRAIC EXPRESSION!

Rings a bell? What is an algebraicexpression? Do you still remember?

Give exam les then.

Warm up your brain cells! Let us begin withexploratory activities that will guide you through the lesson on

first-degree equations and inequalities in one variable.


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Activity 2

Can You Spot the Difference?

Examples of mathematical phrases and mathematical sentences are givenbelow. Study the following and spot the difference.

Mathematical Phrases Mathematical Sentences

1220 81220 b2 42 b yx yx 7

s5 ts 5 43 r 043 c7 17 c

de 4 2

14 de

wv 2 awv 92

What do you observe? How do you compare a mathematical phrasewith amathematical sentence?

Activity 3

Look, Observe and Point Out

The equations below are first-degree equations in one variable.

1x 45 b 072 a 256 c 287 y

The equations below are not first-degree equations in one variable.

12 y 145 rb 072 3 ba 256 4 cc xy 28

Look at the equations above. Observe them carefully and point out theirdifferences. Based on your observation, what are first-degree equations in onevariable?

Which of the following equations are first-degree equations in one variable?

158 x 03 z

2

163 ha 415 r 7t


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FIRM UP Your Understanding

Equations and Inequalities

In Module 2.1, you have learned about algebraic expressions.

The mathematical phrases in Activity 2 are algebraic expressions.

A mathematical phrase is an algebraic expression. It does not express acomplete thought. On the other hand, a mathematical sentencecontains algebraic

expressions together with a relation symbol =, or and it expresses acomplete thought. We recall that these relation symbols are read as follows.

= is equal to or equals< is less than

is less than or equal to> is greater than

is greater than or equal to

Now, let us look at the given mathematical sentences. The first fourmathematical sentences in Activity 2, 81220 , 42 b , yx 7 and ts 5 are

called equations. Can you give your own examples of equations?

An equation is a mathematical sentence that makes use of the symbol =.What do you think does the symbol = imply?

The symbol = implies that the two sides of the equation are equal. Thismeans that whatever is the value of the left side of the equation is also the value of

the right side.

An algebraic expressionis a collection of constants andvariables that are combined using one or more of the four fundamentaloperations namely, addition, subtraction, multiplication and division(except division by zero).

Now lets keep going! Enjoy learning more andmore about first-degree equations and inequalities inone variable .Here are enabling activities that will helpyou.


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This time, let us focus on the last four mathematical sentences in Activity 2,

043 , 17 c ,2

14 de and awv 92 . These mathematical sentences are

called inequalities. Can you give your own examples of inequalities?

How do you then define an inequality?

An inequality is a mathematical sentence that makes use of the relation

symbols or .

What do the symbols < and > imply?

The symbols < and > imply that the left side of the inequality is not equal tothe right side of the inequality. This means further that the symbol < is used whenthe value of the left side of the inequality is less than the value of the right side, whilethe symbol > is used when the value of the left side of the inequality is greater thanthe value of the right side.

The symbol means that the value of the left side of the inequality is either

less than or equal to the value of the right side, while the symbol means that thevalue of the left side of the inequality is either greater than or equal to the value ofthe right side.

Activity 4

Again, go back to each of the given mathematical sentences and tell whether

it is true or false.

Mathematical sentence True or False?

1. 81220 _____2. 42 b _____3. yx 7 _____

4. ts 5 _____5. 043 _____6. 17 c _____

7. 2

1

4 de _____

8. awv 92 _____

If your answer is true for the first mathematical sentence, false for the 5thmathematical sentence, while may be true or false or neither true nor false, for theremaining mathematical sentences, then you are correct.

Sentences 2, 3 4, 6, 7 and 8 may be true or false depending upon the value/sof the variable/s. For example, in the equation 42 b ,

if 2b , then 422 and the equation is true,

but if 1b , then 412 thus, the equation is false.


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Sentences 2, 3, 4, 6, 7 and 8 are examples of open sentences. An opensentenceis an equation or inequality that becomes true or false when the variable isreplaced by a value.

Translating English Statements to Equations or Inequalitiesand Vice Versa

A knowledge of mathematical symbols and their meanings will enable you totranslate verbal sentences into mathematical sentences and vice-versa.

Word/Phrase Symbol

added to, increased by, more than,the sum of, plus

+

subtracted from, decreased by, diminished by, lessthan, the difference, minus -

as much as, of, as many as, the product of

or ( )or sometimes not writtenanymore, i.e., a numberis simply written togetherwith variables

divided by, the quotient of, ratio, over , /, __

is equal to, equals, is the same as =is less than is greater than or equal to, at least

Example 1Translate each of the following into mathematical sentences.

1.1 English sentence: Three times a number is nine.

translation: 3 n = 9

mathematical sentence: 93 n or 93 n or 93 n

We note that the symbol for the operation multiplication may not be writtenanymore.

1.2 English sentence: The sum of a number and seven is twelve.

translation: b + 7 = 12mathematical sentence: 127 b


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1.3 English sentence: The difference between a number and one is eight.

translation: c - 1 = 8mathematical sentence: 81c

Let us see if you can do the same thing in the following sentences. Write thecorresponding symbols below.

1.4 English sentence: A number added to six is greater than two.

translation: ___ ___ ___ ___ ____

mathematical sentence: ______________

1.5 English sentence: Twice a number subtracted by nine is less than five.

translation: ___ ___ ___ ___ ___ ___

mathematical sentence: ________________

Example 2.

Translate each mathematical sentence into an English sentence.

2.1 mathematical sentence: 16xy x

English sentence: The product of x and y is sixteen.

We note that the given mathematical sentence may also be translated as Theproduct of a number and another number is16. or x timesy is equal to16.

2.2 mathematical sentence: 74 a English sentence: The sum of a number and four is greater than or equal

to seven.

Now, let us see if you can translate the given mathematical sentences into anEnglish sentence.

2.3 mathematical sentence: 325 y

English sentence: ____________________________________________ .

2.4 mathematical sentence: 1093 r English sentence: ____________________________________________ .


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Activity 5

Translate each English sentence into a mathematical sentence and identify ifeach is an equation or an inequality.

1. A number, p , minus sixteen is equal to twenty-eight.2. Seven times the sum of negative eight and a number,b , is greater than or

equal to ninety-nine.3. Fifteen is greater than the sum of four and the square of a number, x .4. Seven subtracted from four times a number, a , is less than or equal to the

square of the number, a .5. The quotient when a number, x , is divided by three is equal to negative

eleven.

Differentiating First-Degree Equations fromFirst-degree Inequalities in One Variable

A first-degree equation in one variable is an equation that contains onlyone variable and the variable is raised to the exponent 1. Thus, a first-degreeequation in x is of the form 0 bax , where a is a nonzero real number and b isany real number.

Consider the first set of first-degree equations in one variable that are given inActivity 3.

1x 45 b 072 a 256 c 287 y

If the symbol = is changed to any of the following relation symbols, or , thenwe have first-degree inequalities in one variable. Some possible results are asfollows.

1x 45 b 072 a 256 c 287 y

What is a first-degree inequality in one variable?

A first-degree inequality in one variableis an inequality that contains onlyone variable and the variable is raised to the exponent 1.

A first-degree inequality in x is any of the following forms:0 bax 0 bax 0 bax 0 bax

where a is a nonzero real number and b is any real number.


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Activity 6

1. Give your own examples of first-degree equations in one variable.2. Give examples of first-degree inequalities in one variable.

Activity 7

Identify which of the following is a first-degree equation or inequality in onevariable.

1. 0123 2 xx 6. 1 yx

2. 319 y 7. 4314 x

3. 752

1z 8. 022 cxax

4. 1515 2 ba 9. 23 nm

5. 188

3

b 10. 10023m

Applications of First-Degree Equations and Inequalities

Equations and inequalities are used to model some real-life situations. This issuccessfully done by using your knowledge in translating an English sentence intoan equation or inequality.

Study the following examples.

1. In 1994, twice the population (n ) of a barangay in Bulacan was 50 000. Thisis modeled by an equation that is obtained by translating the sentence Twicen is 50 000. into an equation. Thus, we have 500002 n .

2. The distance, d , that a vehicle travels is computed by multiplying the rate, r,by the time, t , it consumes. In symbols, this is written as rtd . Whatequation represents the time consumed by a plane in travelling a distance of1,468 miles at the rate of 400 mi/hr.?The equation is t400468,1 .

3. Patrick is 4 inches taller than James. The sum of their heights is less than 7feet. Represent this by a first-degree inequality in one variable.

If you use the variable p for Patricks height, then James height is 4p .

(You can also use other variables.) Thus, your final answer must be74 pp .

4. The amount earned by Jonathan is three times the amount earned by Arthur.If you use the variable a to represent the amount earned by Arthur and their

total earnings is at least Php28,000, what first-degree inequality in onevariable will be used to model the situation?


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You should have represented the amount earned by Jonathan as a3 and yourfinal answer must be 000,283 aa .

5. Anikas age is half of Marielles age. Suppose Marielles age is representedby m . What first-degree equation in one variable will represent the sentence,

Ten years from now, their total ages will be 54?

You should have used m2

1for Anikas age. Ten years from now, the

ages of Marielle and Anika should be represented by 10m and 102

1m ,

respectively. Why?

Yes, ten years from now is translated as + 10. Thus, your first-degree

equation must be 54102

110 mm .

Activity 8

Write on equation or inequality to model each situation/problem.

1. Im thinking of a number, n . If 12 is added to it, the result is 79.

2. If my mother would increase my weekly allowance by Php60, it would bemore than Php310. If a represents my weekly allowance, write aninequality to find the possible amounts for my weekly allowance.

3. Glenn weighs 7 kg more than his brother Raymond. What first-degreeinequality represents the statement, Together they weigh at least 124 kg.,if r represents Raymonds weight?

4. The width of a rectangle is 43 dm. Find the length which will make its areaat most 3096 square dm. Represent the length by the variable l .

5. Mang Jose earns Php65 an hour. How long must he work to earn morethan P520? Suppose t represents the number of hours that Mang Jose

works, write an inequality to solve this problem.


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DEEPEN Your Understanding

Activity 9

A. Tell whether each of the following is a mathematical phrase or amathematical sentence.

1. 22

1x 6. 14212 x

2. 5223 xx 7. 224 dc 3. 75 y 8. x10

4.

y

y1

9. 13511 xx

5. yy 52 10. 132 nn

B. Classify the following mathematical sentences as true, false or open.

11. One kilometer is equal to 1,000 meters.12. It is the worlds largest archipelago.13. 1495 14. 2049 x 15. 2115

C. Fill in the box with the relation symbol =, or to make it a truestatement.

16. 53 19

17. 8 372

18. 2339 1526

19. 6x 13 , if x is replaced by 7

20. m2 96 , if m is 1

Getready to take on more challenges to yourmathematical thinking and reasoning in relation tothe lessons. Do the following activities to furtherenhance your skills.


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Activity 10

A. Translate each English sentence into a mathematical sentence.

1. A number,m , added to six is equal to two.2. A number, p , minus 16 is equal to 38.

3. The difference between a4 and 7 is less than 6.4. Seven times the sum of 8 and a number, b , is greater than or equal to

10.5. Six times a number, y , less four is equal to eight.

B. Translate the given mathematical sentence into an English sentence.

6. 952 x

7. 1839 x 8. 4162 x

9. 812 x 10. 1634 m

Activity 11

Determine whether each of the following is an example of a first-degreeequation or a first-degree inequality in one variable. Explain your answer.

1. 01x

2. 39

2

y

3. 0144 2 aa 4. 25 nm 5. c39

Activity 12

Read the following situations and do what is required.

1. Kristines weight is 2 lbs less than the weight, w of Ronald. Write a first-degreeequation in one variable that represents the sentence The sum oftheweights of Kristine and Ronald is 210 lbs.

2. The perimeter, p of a rectangle with length, l and width, w is given by the

formula wlp 22 . The length of a rectangular table is 1 m more than its

width. What is the first-degree equation in one variable that relates theperimeter and width of the table if the perimeter is 6 m?


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3. Belle sold 20 more magazines than Chris. If you use the variable, c torepresent the number of magazines sold by Chris, what first-degree equation inone variable represents the sentence Five times the total number of magazinessold by Belle and Chris is ten more than seven times the number of magazinessold by Belle.?

4. The number of Php10 coins is 17 decreased by the number of Php5 coins. If thevariable f is used to represent the number of P5 coins, how will you represent

the following?

4.1 the number of Php10 coins in terms of f

4.2 the value of Php5 coins4.3 the value of Php10 coins4.4 the first-degree mathematical sentence in one variable for The value of all

coins is at most Php110.

5. Running, Bryan covered a distance, b while Louie covered a third of this. Writethe first-degree equation that models the statement, The distance covered byBryan is four times the distance covered by Louie.

TRANSFER Your Understanding

Activity 13

A. Based on what you have learned, think of real-life situations that canbe modelled by first-degree equations and inequalities in one variable, similar

to those in Activity 12 . Compile your work in your portfolio.

B.Write a journal about your views regarding the importance of

first-degree equations and inequalities in one variable in modelling real-liferelationships of various quantities.

Its time to demonstrate what you havelearned. Do the following activities and compileyour work in your portfolio.


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Answers KeyModule 2.5: First-Degree Equations and Inequalities in One Variable

Activity 1

An algebraic expressionis an expression composed of constants, variables,grouping symbols, and operation symbols.

Some examples are: yx 23 , 26m , yx 3 .(Given examples may vary.)

Activity 2

Difference: The expressions under the column of mathematical sentencesmade use of the relation symbols such as , =, , >, and


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Activity 6

(Answers may vary.)

Activity 7

1. not2. first-degree equation in one variable3. first-degree equation in one variable4. not5. first-degree inequality in one variable6. first-degree equation in one variable7. first-degree inequality in one variable8. not9. first-degree inequality in one variable

10. first-degree equation in one variable

Activity 8

1. 7912 n 2. 31060 a 3. 1247 rr

4. 309643 l 5. 52065 t

Activity 9

A.1. mathematical phrase 6. mathematical sentence2. mathematical sentence 7. mathematical phrase3. mathematical sentence 8. mathematical phrase4. mathematical phrase 9. mathematical sentence5. mathematical phrase 10. mathematical sentence

B.

11. true 14. open12. open 15. false13. true

C.16. = 19. =17. = 20.


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Activity 10

A.

1. 26 m 4. 1087 b 2. 3816 p 5. 846 y

3. 674 a

B.6. The sum of twice a number and five is equal to nine.

or Twice a number increased by five is equal to nine.7. Thrice a number added to nine is equal to eighteen.8. Twice a number less sixteen is less than or equal to four.

or Sixteen subtracted from twice a number is less than or equal to four.9. Twice the sum of a number and one is equal to eight.

10. Three subtracted from four times a number is greater than or equal to

sixteen.

Activity 11

1. first-degree equation in one variableThe only variable used is x and its exponent is one.

2. notThe exponent of the variable y is two.

3. notThe degree is two because the exponent of the variable a is two.

4. notThere are two variables, m and n .

5. first-degree equation in one variableThere is only one variable, c with an exponent of one.

Activity 12

1. 2102 ww 2. ww 2126

3. 10207205 ccc 4. 4.1. f17 4.2. f5

4.3. f1710 4.4. 11017105 ff

5.

bb

3

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Math 1 Module 2