[email protected] MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

[email protected] • MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

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Chabot Mathematics§1.1 §1.1

ExpressionsExpressions& Real No.s& Real No.s



Basic TerminologyBasic Terminology

A LETTER that can be any one of various numbers is called a VARIABLE.

If a LETTER always represents a particular number that NEVER CHANGES, it is called a CONSTANT

A & B are CONSTANTS



Algebraic ExpressionsAlgebraic Expressions

An ALGEBRAIC EXPRESSION consists of variables, numbers, and Math-Operation signs.• Some

Examples , 2 2 , .4

yt l w m x b

When an equal sign is placed between two expressions, an EQUATION is formed →

374

2 ty

cmEbxmy



Translate: English → AlgebraTranslate: English → Algebra

“Word Problems” must be stated in ALGEBRAIC form using Key Words

per of less than more than

ratio twicedecreased byincreased by

quotient of times minus plus

divided byproduct ofdifference of sum of

divide multiply subtract add

DivisionMultiplicationSubtractionAddition



Example Example Translation Translation

Translate this Expression:

Eight more than twice the product of 5 and an Unknown number

SOLUTION• LET n ≡ the UNknown Number

8 2 5 n

Eight more than twice the product of 5 and a number.



Evaluate Algebraic ExpressionsEvaluate Algebraic Expressions

When we REPLACE A VARIABLE with a number, we are SUBSTITUTING for the variable. The calculation that follows is

called EVALUATING the expression• Note: the normal result for a “Evaluation”

is usually a SINGLE NUMBER with NO LETTERS



Example Example Evaluating Evaluating

Evaluate

8 for 2, 7, and 3.xz y x y z

SOLUTION

8xz − y = 8·2·3 − 7

= 41

= 48 − 7

Substituting

Multiplying

Subtracting



Exponential NotationExponential Notation

The expression an, in which n is a counting number (1, 2, 3, etc.) means

n factors

In an, a is called the base and n is called the exponent, or power. When no exponent appears, it is assumed to be 1. Thus a1 = a.

a a a a a



Order of Operations (PEMDAS)Order of Operations (PEMDAS)

Perform operations in this order:1. Grouping symbols: parentheses ( ),

brackets [ ], braces { }, absolute value | |, and radicals √

2. Exponents from left to right, in order as they occur.

3. Multiplication/Division from left to right, in order as they occur.

4. Addition/Subtraction from left to right, in order as they occur.



Example Example Order of Ops Order of Ops

Evaluate 2 22 3 12 for 2.x x x

SOLUTION

2(x + 3)2 – 12 x2

Substituting

Simplifying 52 and 22

Multiplying and Dividing

Subtracting

= 2(2 + 3)2 – 12 22

2 22 5 12 2 2 25 12 4

Working within parentheses

50 3 47



FormulasFormulas

A FORMULA is an equation that uses letters to represent a RELATIONSHIP between two or more quantities.

Example The area, A, of a circle of radius r is given by the formula:

2rA r



Example Example Temp Conversion Temp Conversion

The formula for converting the temperature in degrees Celsius (C) to degrees Fahrenheit (F)

F 9

5C 32.

Use the formula to convert 86F to the Celsius form

Substitute 86 for F 86 9

5C 32

Solve for C 5 86 59

5C 32



Temperature Conversion cont.Temperature Conversion cont.

Solve for C. 5 86 59

5C 32

430 9C 160

270 9C

270

9C

30 C

Thus 30 °C is the Equivalent of 86 °F



Mathematical ModelingMathematical Modeling

A mathematical model can be a formula, or set of formulas, developed to represent a real-world situation.



Example Example Math Model Math Model

Mei-Li is 5ft 7in tall with a Body Mass Index (BMI) of approximately 23.5. What is her weight?

SOLUTION1. Familiarize. The body mass index I

depends on a person’s height and weight. The BMI formula:

2

704.5WI

H

– where W is the weight in pounds and H is the height in inches.



BMI Example contBMI Example cont

2. Translate. Solve the formula for W:

2

704.5WI

H

2 704.5IH W2

704.5

IHW

3. Carry Out

5 ft 7 in. = 67 in.

2 223.5(67)

704.5 704.5

IHW

150W



BMI Example contBMI Example cont

4. Check 5. State Answer

2

704.5WI

H

2

704.5(150)

67I

23.5I

Mei-Li weighs about 150 Pounds



Set of Real NumbersSet of Real Numbers

SET ≡ a Collection of Objects

Braces are used to indicate a set. For example, the set containing the numbers 1, 2, 3, and 4 can be written {1, 2, 3, 4}.• The numbers 1, 2, 3, and 4 are called the

members or elements of this set.



Set NotationSet Notation

Roster notation: {2, 4, 6, 8}

Set-builder notation: {x | x is an even number between 1 and 9}

“The set of all x such that x is an even number between 1 and 9”



Sets of Real NumbersSets of Real Numbers

Natural Numbers (Counting Numbers)• Numbers used for counting: {1, 2, 3,…}

Whole Numbers• The set of natural numbers with 0 included:

{0, 1, 2, 3,…}

Integers• The set of all whole numbers AND their

opposites: {…,−3, −2, −1, 0, 1, 2, 3,…}



Sets of Real Numbers, Sets of Real Numbers, xx

Rational Numbers (Integer Fractions)• Maybe expressed as a FRACTION of

two INTEGERS– Terminating Decimals (e.g.; 7/16)

– Repeating NonTerminating Decimals (e.g., 2/7)

Irrational Numbers• Can NOT be expressed as a Fraction of

two Integers– NONterminating, NONreapeating decimals

(e.g., π = 3.1459………………………)



4 77 85, ,0, ,9.45, 8.734

Irrational numbers:

10,

3,

,

15,

7.161161116...

Integers: 5 192

8 3 7, , ,3.4

19, 3, 1

Zero: 0

Real Numbers:

2319, 10,0, , ,17.8

Rational Numbers:

19, 1,0,5,23

Rational numbers that are not integers:

NegativeIntegers:

Whole numbers:0, 1, 2, 3, 29

Positive integers or natural numbers:1, 2, 3, 29

Real N

o. F

amily T

reeR

eal No

. Fam

ily Tree



Real Number NestReal Number Nest



Real Number LineReal Number Line

An Infinite line whose points have been assigned number-coordinates is called the real number line



3 Sectors of the Number Line3 Sectors of the Number Line

1. The negative real numbers are the CoOrds to the left of the origin O

2. The real number zero is the CoOrd of the origin O

3. The positive real numbers are the CoOrds to the right of the origin O



InEqualitiesInEqualities

< means “is less than” ≤ means “is less than or equal to” > means “is greater than” ≥ means “is greater than or equal to” For any two numbers

on a number line, the one to the left is said to be less than the one to the right.

–1 0 1 2 3

–1 < 3 since –1 is to the left of 3




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Chabot Mathematics

§1.2 §1.2 Operations Operations with Real with Real

No.sNo.s



Absolute ValueAbsolute Value

The ABSOLUTE VALUE of a number is its distance from zero on a number line.

The symbol |x| to represents the absolute value of a number x.

Example |5| = |−5| = 5

5 units from 0 5 units from 0

• “Distance” is ALWAYS Positive



Find Absolute ValueFind Absolute Value Finding Absolute Value

• If a number is negative, make it positive

• If a number is positive or zero, leave it alone

Example Find the absolute value of each number.

• a) |−4.5| b) |0| c) |−3|

Solution

• a) |−4.5| The dist of −4.5 from 0 is 4.5, so |−4.5| = 4.5

• b) |0| The distance of 0 from 0 is 0, so |0| = 0.

• c) |−3| The distance of –3 from 0 is 3, so |−3| = 3



Real No. Addition RulesReal No. Addition Rules1. Positive numbers: Add the numbers.

The result is positive.2. Negative numbers: Add absolute values.

Make the answer negative.3. A positive and a negative number: Subtract the

smaller absolute value from the larger. Then:a) If the positive number has the greater absolute value,

make the answer positive.b) If the negative number has the greater absolute

value, make the answer negative.c) If the numbers have the same absolute value,

the answer is 0.

4. One number is zero: The sum is the other number



Inverse Property of AdditionInverse Property of Addition

For any real number a, the opposite, or additive inverse, of a, (which is −a) is such that

a + (−a) = −a + a = 0 Example Find the opposite, or

additive inverse: a) 8 b) −13 Solution

• a) 8 −(8) = −8

• b) −13 −(−13) = 13



SubtractionSubtraction

Subtraction ≡ The difference a − b is the unique number c for which a = b + c. • That is, a − b = c if c is a number

such that a = b + c

Subtracting by Adding the Opposite• For any real numbers a and b

a − b = a + (−b) – We can subtract by adding the opposite

(additive inverse) of the number being subtracted (the Subtrahend)



Real No. MultiplicationReal No. Multiplication

The Product of a Positive and a Negative Number• To multiply a positive number and a

negative number, multiply their absolute values. The answer is negative: 3(−2) = −6

The Product of Two Negative Numbers• To multiply two negative numbers, multiply

their absolute values. The answer is positive: (−13)(−11) = +143



Real No. DivisionReal No. Division

The quotient a b or a/b where b ≠ 0, is that unique real number c for which a = b • c.

Example Divide 8

64

5

45

)) ba

SOLUTION

95

45

)a Because −5 · (−9 ) = 45

Because −8 · (8) = −6488

64

)b



Multiply & Divide RulesMultiply & Divide Rules To multiply or divide two real numbers:

1. Multiply or divide the absolute values.2. If the signs are the same, then the

answer is positive.3. If the signs are different, then the answer

is negative. DIVISION by ZERO

• NEVER divide by ZERO. If asked to divide a nonzero number by zero, we say that the answer is UNDEFINED. If asked to divide 0 by 0, we say that the answer is INDETERMINATE.



Real Number PropertiesReal Number Properties

COMMUTATIVE property of addition and multiplication

a + b = b + a and ab = ba ASSOCIATIVE property

a + (b + c) = (a + b) + c and a(bc) = (ab)c

DISTRIBUTIVE property

a(b + c) = ab + ac



Real Number PropertiesReal Number Properties ADDITIVE IDENTITY property

a + 0 = 0 + a = a ADDITIVE INVERSE property

−a + a = a +(−a) = 0 MULTIPLICATIVE IDENTITY property

a • 1 = 1 • a = a MULTIPLICATIVE INVERSE property

a •(1/a) = (1/a) • a = 1 (a ≠ 0)



Simplify ExpressionsSimplify Expressions A TERM is a number, a variable, a

product of numbers and/or variables, or a product or quotient of two numbers and/or variables.

Terms are SEPARATED by ADDITION signs. If there are SUBTRACTION signs, we can find an equivalent expression that uses addition signs.

COLLECTING LIKE TERMS is based on the DISTRIBUTIVE law



Like (or Similar) TermsLike (or Similar) Terms Terms in which the variable factors are

exactlyexactly the same, such as 9x and −5x, are called like, or similar terms.

Like Terms UNlike Terms

7x and 8x 8y and 9y2

3xy and 9xy 5ab and 4ab2



Example Example Combine Terms Combine Terms

a) 7x + 3x b) 4a + 5b + 2 + a − 6 − 5b

SOLUTIONa) 7x + 3x = (7 + 3)x = 10x

b) 4a + 5b + 2 + a − 6 − 5b

= 4a + 5b + 2 + a + (−6) + (−5b)

= 4a + a + 5b + (−5b) + 2 + (−6)

= [4a + a] + [5b + (−5b)] + [2 + (−6)]

= 5a + 0 + (−4)

= 5a − 4



Example Example Simplify Simplify

Remove parentheses and simplify

[6(m + 3) – 5m] – [4(n + 5) – 8(n – 4)]

SOLUTION[6(m + 3) – 5m] – [4(n + 5) – 8(n – 4)]

= [6m + 18 – 5m] – [4n + 20 – 8n + 32] Distribute

= [m + 18] – [–4n + 52] Collect like terms within brackets

= m + 18 + 4n – 52 Removing brackets

= m + 4n – 34 Collecting like terms



TERMSTERMS ≠ ≠ factorsfactors

Factors are the “pieces” of a Multiplication Chain; e.g., if

• Then y has four factors: 7, u, v, w

TERMS are the pieces of an ADDITION Chain

• Then z has Three TERMS: 7a, 3b, −5c

cbacbaz 537537

wvuyuvwy 7or7




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§1.3 §1.3 GraphingGraphingEquationsEquations



Points and Ordered-PairsPoints and Ordered-Pairs

To graph, or plot, points we use two perpendicular number lines called axes. The point at which the axes cross is called the origin. Arrows on the axes indicate the positive directions

Consider the pair (2, 3). The numbers in such a pair are called the coordinates. The first coordinate, x, in this case is 2 and the second, y, coordinate is 3.



Plot-Pt using Ordered PairPlot-Pt using Ordered Pair

To plot the point (2, 3) we start at the origin, move horizontally to the 2, move up vertically 3 units, and then make a “dot”• x = 2

• y = 3

(2, 3)



Example Example Plot the point ( Plot the point (−−4,3)4,3)

Starting at the origin, we move 4 units in the negative horizontal direction. The second number, 3, is positive, so we move 3 units in the positive vertical direction (up)• x = −4; y = 3

4 units left

3 u

nit

s u

p



Example Example Read XY-Plot Read XY-Plot Find the coordinates of pts A, B, C, D, E, F, G

AB

C

D

E

F

G

• Solution: Point A is 5 units to the right of the origin and 3 units above the origin. Its coordinates are (5, 3). The other coordinates are as follows: – B: (–2, 4)

– C: (–3, –4)

– D: (3, –2)

– E: (2, 3)

– F: (–3, 0)

– G: (0, 2)



XY QuadrantsXY Quadrants

The horizontal and vertical axes divide the plotting plane into four regions, or quadrants



Graphing EquationsGraphing Equations Definitions

• An ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true.

• An ordered pair that satisfies an equation is called a solution of the equation

• Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable.



Eqn Graph Eqn Graph Bottom Line Bottom Line

ANY and ALL points (ordered pairs) on a Math Graph are SOLUTIONS to the Equation that generated the Graph



Graph of an EquationGraph of an Equation

The graph of an equation in two variables, such as x and y, is the set of all ordered pairs (a, b) in the coordinate plane that satisfy the equation

y = 2x + 6 −2x + y = 6

2x − y + 6 = 0



Graphing by Plotting PointsGraphing by Plotting Points

Graph y = x2 – 3 SOLUTION → Make “T-table”

(–3, 6)y = (–3)2 – 3 = 9 – 3 = 6–3

(x, y)y = x2 – 3x

X col

Pick x

Calc y Ordered Pair

(–3, 6) is a solutionto y = x2 – 3

[ 6 = 32 – 3 ]



Graph by Pt-Plot Graph by Pt-Plot yy = = xx22 – 3 – 3

Pick “enough” x’s for T-table

(3, 6)y = 32 – 3 = 9 – 3 = 63

(2, 1)y = 22 – 3 = 4 – 3 = 12

(1, –2)y = 12 – 3 = 1 – 3 = –21

(0, –3)y = 02 – 3 = 0 – 3 = –30

(–1, –2)y = (–1)2 – 3 = 1 – 3 = –2–1

(–2, –1)y = (–2)2 – 3 = 4 – 3 = 1–2

(–3, 6)y = (–3)2 – 3 = 9 – 3 = 6–3

(x, y)y = x2 – 3x



Graph by Pt-Plot Graph by Pt-Plot yy = = xx22 – 3 – 3

Plot (x,y) Points listed in T-table and connect the dots to complete the plot• Note that most

Graphs are “curves” so connect dots with curved lines



Graph by Pt-Plot: y = Graph by Pt-Plot: y = xx22 – 2 – 2xx – 6 – 6 Construct T-table

x y (x, y)

3 9 (3, 9)

2 2 (2, 2)

0 6 (0, 6)

1 7 (1, 7)

2 6 (2, 6)

3 3 (3, 3)

4 2 (4, 2)

5 9 (5, 9)

Plot-Pts & Connect-Dots



GRAPH BY PLOTTING POINTSGRAPH BY PLOTTING POINTS Step1. Make a representative

T-table of solutions of the equation.

Step 2. Plot the solutions as ordered pairs in the Cartesian coordinate plane.

Step 3. Connect the solutions in Step 2 by a smooth curve



Domain & Range by GraphingDomain & Range by Graphing

Graph y = x2. Then State the Domain & Range of the equation

Select integers for x, starting with −2 and ending with +2. The T-table:

x 2xy Ordered Pair yx,

2 42 2 y 4,2

1 11 2 y 1,1

0 002 y 0,0

1 112 y 1,1

2 422 y 4,2



Example Example Domain & Range Domain & Range

Now Plot the Five Points and connect them with a smooth Curve

-2

-1

0

1

2

3

4

5

6

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

M55_§JBerland_Graphs_0806.xls

x

y

(−2,4) (2,4)

(−1,1) (1,1)

(0,0)




The DOMAIN of a function is the set of ALL first (or “x”) components of the Ordered Pairs that appear on the Graph

Projecting on the x-axis ALL the x-components of ALL POSSIBLE ordered pairs displays the DOMAIN of the function just plotted




Domain of y = x2 Graphically

-2

-1

0

1

2

3

4

5

6

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


x

y

This Projection Pattern Reveals a Domain of

number real a isxx




The RANGE of a function is the set of all second (or “y”) components of the ordered pairs. The projection of the graph onto the y-axis shows the range -2

-1

0

1

2

3

4

5

6

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


x

y




The projection of the graph onto the y-axis is the interval of the y-axis at the origin or higher, so the range is

-2

-1

0

1

2

3

4

5

6

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


x

y

0yy




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§1.4 Solve§1.4 SolveLinear Linear EqnsEqns



Solution For an EquationSolution For an Equation

Any NUMBER-REPLACEMENT for the VARIABLE that makes an equation true is called a SOLUTION of the equation. To Solve an equation means to find

ALL of its Solutions.



Equivalence and Addition Principle Equivalence and Addition Principle

Equivalent Equations• Equations with the SAME

SOLUTIONS are called EQUIVALENT equations

Addition Principle• For any real numbers u, v, and w,

u = v is equivalent to u + w = v + w– e.g.; u = 3, v = 3, w = 7 →

3 = 3 and 3+7 = 3+7



Multiplication PrincipleMultiplication Principle

The Multiplication Principle: For any real numbers r, s, and t with t ≠ 0, r = s is equivalent to r•t = s•t

Example• Solve for x

154

3x

Solution

3

415

4

3

3

4x Multiply Both

Sides by 4/3

203

4351

x

20x



Linear EquationsLinear Equations

A linear equation in one variable, such as x, is an equation that can be written in the standard form

where a and b are real numbers with a ≠ 0

0bax



Solution to Linear EquationsSolution to Linear Equations Procedure for solving linear equations

in one variable

1. Eliminate Fractions: if Needed, Clear Fractions by Multiplying both sides of the equation by the least common denominator (LCD) of all the fractions

2. Simplify: Simplify both sides of the equation by removing parentheses and other grouping symbols (if any) and combining like terms



Solution to Linear EquationsSolution to Linear Equations

3. Isolate the Variable Term: Add appropriate expressions to both sides, so that when both sides are simplified, the terms containing the VARIABLE are on ONE SIDE and all constant terms are on the other side.

4. Combine Terms: Combine terms containing the variable to obtain one term that contains the variable as a factor.



Solution to Linear EquationsSolution to Linear Equations

5. Isolate the Variable: Divide both sides by the coefficient of the variable to obtain the solution.

6. Check the Solution: Substitute the solution into the original equation



Example Example Solve Linear Eqn Solve Linear Eqn

Solve for x: 6x 3x 2 x 2 11.

SOLUTION (No Fractions to clear)

6x 3x 2 x 2 11

6x 3x 2x 4 11

6x 3x 2x 4 11

5x 4 11

Step 2





SOLUTION

Step 3 5x 4 4 11 4

Step 4 5x 15

5x

5

15

5x 3

Step 5





SOLUTION

6 3 3 3 2 3 2 11

18 9 2 11

18 7 11

Step 6 Check x = 3:

State The Solution is x = 3




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Chabot Mathematics

§1.5 §1.5 Problem Problem SolvingSolving



Mathematical ModelMathematical Model A mathematical model is an

equation or inequality that describes a real situation. Models for many APPLIED

(or “Word”) problems already exist and are called FORMULAS A FORMULA is a mathematical

equation in which variables are used to describe a relationship



Formula Describes RelationshipFormula Describes Relationship

Relationship Mathematical Formula

Perimeter of a triangle:

a

b

ch

Area of a triangle:



Example Example Geometry of Cone Geometry of Cone

Relationship Mathematical Formulae

h

r

Volume of a cone:

Surface area of a cone:

(slant-sides + bottom)



Example Example °F ↔ °C °F ↔ °C

Relationship Mathematical Formulae

Celsius to Fahrenheit:

Fahrenheit to Celsius:

Celsius Fahrenheit



Example Example Mixtures Mixtures

Relationship Mathematical Formula

Percent Acid, P:

Base

Acid



Solving a FormulaSolving a Formula

Sometimes the formula is solved for a Different variable than the one we need

Example A mathematical model tell us that voltage, V, in a circuit is equal to current, I, times resistance, R: V = I R

To determine the amount of resistance in a circuit, it would help to first solve the formula for R.



Example Example Solve Solve VV = = IRIR for for RR

We solve this formula for R by • treating V and I as CONSTANTS

(having fixed values)

• treating R as the only variable.

Begin by writing the formula so that the variable for which we are solving, R, is on the left side.

I R = V



Example Example Solve Solve VV = = IRIR for for RR

Finally, use Algebra properties to isolate the variable R.

Divide both sides by I.

VIR

I

V

I

IR

I

VR Isolated R → Done



Example Example Trapezoid Base, B Trapezoid Base, B

B

b

h

This formula gives the relationship between the height, h, and two bases, B and b, of a trapezoid and its area, A.




Mult. Prop. of Equality.

Assoc. Prop.

Inverse Prop.

Identity Prop.




Add. Prop. Of Equality

Divide by h.

Distributive Prop.

Inverse Prop.



Example Example Prismatic Volume Prismatic Volume

h

b

l

The volume of a triangular prism is given by:

If the volume of a triangular cylinder is 880 cm3, the base is 10 cm, and the length is 22 cm, then find the height.




First, solve the equation for h.




Second, find the height, h, by substituting the given values of V, b, and l into this formula:

State The height of the triangular prism is 8 cm



Example Example % from a Pie Chart % from a Pie Chart

The pie chart shown at right represents the distribution of grades in MTH2 (Calculus) last year. Use the information in the chart to estimate how many B’s will be given in a new class of size 70 students.

F 8%

A 16%

B 28%

C 36%

D 12%



Example Example % from a Pie Chart % from a Pie Chart According to the chart, 28%

OF the students should get a grade of B. Let x represent the number of students getting a B.

F 8%

A 16%

B 28%

C 36%

D 12%



WhiteBoard WorkWhiteBoard Work

Problems From §1.n Exercise Sets• NONE Today → Lecture PPT took Entire

Class Time

Identity Symbol • Difference of Two Squares Identity

srsrsr 22



All Done for TodayAll Done for Today

The“Defined as”

Symbol

C)( 12-Carbon of

12g EXACTLYin

Atoms ofNumber

10023.6

12

23




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Chabot Mathematics

AppendiAppendixx

–



Set Membership NotationSet Membership Notation

BA

BBA

BA

BABA

AaAa

AaAa

in not is ofelement oneleast at that meanswhich

,"set ofsubset anot isA set " read is

in is in element every meanswhich

,"set ofsubset a is set " read is

"set ofmember anot is " read is

set ofmember a is " read is

"



Tool For XY GraphingTool For XY Graphing Called “ Engineering

Computation Pad”• Light Green

Backgound

• Tremendous Help with Graphing and Sketching

• Available in Chabot College Book Store

• I use it for ALL my Hand-Work

Graph on this side!

[email protected] MTH55_Lec-01_sec1_1-5_Alge_RealNos_Graphs_ProbSolv.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical.

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