Basic Concepts of Algebra

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Chapter 1

• Basic Concepts of Algebra

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LANGUAGE OF ALGEBRA
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SET a collection or group of, things, objects,
numbers, etc.
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INFINITE SET a set whose members cannot be
counted.
If A 1, 2, 3, 4, 5, then A is infinite
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FINITE SET a set whose members can be counted.
If A e, f, g, h, i, j then A is finite and
contains six elements
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SUBSET all members of a set are members of
another set
If A e, f, g, h, i, j and B e, i , then B?A
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EMPTY SET or NULL SET a set having no
elements.

A or B are empty sets or null sets
written as ?
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• 1-1
• Real Numbers and Their Graphs

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• Real Numbers

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NATURAL NUMBERS - set of counting numbers
1, 2, 3, 4, 5, 6, 7, 8
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WHOLE NUMBERS - set of counting numbers plus zero
0, 1, 2, 3, 4, 5, 6, 7, 8
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INTEGERS - set of the whole numbers plus their
opposites
, -3, -2, -1, 0, 1, 2, 3,
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RATIONAL NUMBERS - numbers that can be
expressed as a ratio of two integers a and b and
includes fractions, repeating decimals, and
terminating decimals
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EXAMPLES OF RATIONAL NUMBERS
½, ¾, ¼, - ½, -¾, -¼, .05 .76, .333, .666, etc.
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IRRATIONAL NUMBERS - numbers that cannot be
expressed as a ratio of two integers a and b and
can still be designated on a number line
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EXAMPLES OF IRRATIONAL NUMBERS
?, ?6, -?29, 8.11211121114, etc .
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1. Each point on a number line is paired with
   exactly one real number, called the coordinate of
   the point.
2. Each real number is paired with exactly one point
   on the line, called the graph of the number

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• 1-2
• Simplifying Expressions

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• Definitions

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NUMERICAL EXPRESSION or NUMERAL a symbol or
group of symbols used to represent a number

3 x 4 5 5 2 15 - 3 24 2 12
2 x 6
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VALUE of a Numerical Expression The number
represented by the expression

Twelve is the value of 3 x 4 5 5 2
15 - 3 24 2 12 2 x 6
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EQUATION a sentence formed by placing an equals
sign between two expressions, called the sides
of the equation. The equation is a true
statement if both sides have the same value.

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EXAMPLES OF EQUATIONS
-6 10 6 2 or 4x 3 19
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INEQUALITY SYMBOL One of the symbols lt - less
than gt greater than ? - does not equal - less
than or equal to - greater than or equal to

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INEQUALITY a sentence formed by placing an
inequality symbol between two expressions, called
the sides of the inequality -3 gt -5 -3 lt - 0.3

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SUM the result of adding numbers, called the
terms of the sum 6 15 21 10 2 12

sum
terms
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DIFFERENCE the result of subtracting one number
from another 8 6 2 10 - 2 8

difference
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PRODUCT the result of multiplying numbers, called
the factors of the product 6 x 15 80 10 2 20

product
factors
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QUOTIENT the result of dividing one number by
another 35 7 5 10 2 5

quotient
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POWER, BASE, and EXPONENT A power is a product of
equal factors. The repeated factor is the base.
A positive exponent tells the number of times the
base occurs as a factor.

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EXAMPLES OF POWER, BASE, and EXPONENT Let the
base be 3. First power 3 31 Second power 3 x
3 32 Third power 3 x 3 x 3 33 Exponent is
1,2,3
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GROUPING SYMBOLS Pairs of parentheses ( ),
brackets , braces , or a bar used to
enclose part of an expression that represents a
single number. 3 4(2 x 6) -22 2
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VARIABLE a symbol, usually a letter, used to
represent any member of a given set, called the
domain or replacement set, of the variable a, x,
or y

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EXAMPLES OF VARIABLES If the domain of x is
0,1,2,3, we write x ? 0,1,2,3
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VALUE of a Variable - the members of the domain
of the variable. If the domain of a is the set
of positive integers, then a can have these
values 1,2,3,4,

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Algebraic Expression a numerical expression a
variable or a sum, difference, product, or
quotient that contains one or more variables

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EXAMPLES OF ALGEBRAIC EXPRESSIONS 24 3 x
y2 2y 6 a b 2c2d 4 c
d
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SUBSTITUTION PRINCIPLE An expression may be
replaced by another expression that has the same
value.
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ORDER OF OPERATIONS

1. Grouping symbols
2. Simplify powers
3. Perform multiplications and divisions in order
   from left to right. and

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ORDER OF OPERATIONS

1. Perform additions and subtractions in order from
   left to right
2. Simplify the expression within each grouping
   symbol, working outward from the innermost
   grouping

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DEFINITION of ABSOLUTE VALUE

• For each real number a,
• l a l a if a gt0
• 0 if a 0
• - a if a lt 0

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• 1-3
• Basic Properties of Real Numbers

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• Properties of Equality

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• Reflexive Property - a a
• Symmetric Property - If a b, then b a
• Transitive Property - If a b, and b c, then a
  c

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• Addition Property - If a b, then a c b c
  and c a c b
• Multiplication Property -If a b, then ac bc
  and ca cb

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• Properties of Real Numbers

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CLOSURE PROPERTIES

• a b and ab are unique
• 7 5 12
• 7 x 5 35

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COMMUTATIVE PROPERTIES

a b b a ab ba
2 6 6 2 2 x 6 6 x 2
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ASSOCIATIVE PROPERTIES

(a b) c a (b c) (ab)c a(bc)
(5 15) 20 5 (15 20) (515)20 5(1520)
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IDENTITY PROPERTIES
There are unique real numbers 0 and 1 (1?0) such
that a 0 0 a a a 1 1 a

-3 0 0 -3 -3 3 x 1 1 x 3 3
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INVERSE PROPERTIES

• PROPERTY OF OPPOSITES
• For each a, there is a unique real number a
  such that
• a (-a) 0 and (-a) a 0 (-a is called
  the opposite or additive inverse of a

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INVERSE PROPERTIES

• PROPERTY OF RECIPROCALS
• For each a except 0, there is a unique real
  number 1/a such that
• a (1/a) 1 and (1/a) a 1 (1/a is
  called the reciprocal or multiplicative inverse
  of a

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DISTRIBUTIVE PROPERTY

a(b c) ab ac (b c)a ba ca
5(12 3) 512 5 3 75 (12 3)5 12 5
3 5 75
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• 1-4
• Sums and Differences

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• Rules for Addition

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For real numbers a and b

• If a and b are negative numbers, then a b is
  negative and a b -(lal lbl)
• -5 (-9) - (l-5l l-9l) -14

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For real numbers a and b

• If a is a positive number, b is a negative
  number, and lal is greater than lbl, then a b
  is a positive number and a b lal lbl
  9 (-5) l9l l-5l 4

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For real numbers a and b

• If a is a positive number, b is a negative
  number, and lal is less than lbl, then a b is a
  negative number and a b -lbl lal
  5 (-9) -l-9l l5l -4

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DEFINITION of SUBTRACTION

• For all real number a and b,
• a b a (-b)
• To subtract any real number, add its opposite

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DISTRIBUTIVE PROPERTY

• For all real number a ,b, and c
• a(b - c) ab ac
• and
• (b c)a ba - ca

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• 1-5
• Products

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MULTIPLICATIVE PROPERTY OF 0

For every real number a, a 0 0 and 0 a 0
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MULTIPLICATIVE PROPERTY OF -1

For every real number a, a(-1) -a and (-1)a
-a
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• Rules for Multiplication

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• The product of two positive numbers or two
  negative numbers is a positive number.
• (5)(9) 45 or (-5)(-9) 45

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• The product of a positive number and a negative
  number is a negative number.
• (-5)(9) -45 or (5)(-9) -45

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• The absolute value of the product of two or more
  numbers is the product of their absolute values
• l(-5)(9)l l-5l l9l 45

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PROPERTY of the OPPOSITE of a PRODUCT

For all real number a and b, -ab (-a)b and -ab
a(-b)
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PROPERTY of the OPPOSITE of a SUM

For all real number a and b, -(a b) (-a)
(-b)
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• 1-6
• Quotients

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DEFINITION OF DIVISION

• The quotient a divided by b is written a/b or
  ab. For every real number a and nonzero real
  number b,
• a/b a1/b, or ab a1/b

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DEFINITION OF DIVISION

• To divide by any nonzero number, multiply by its
  reciprocal. Since 0 has no reciprocal, division
  by 0 is not defined.

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• Rules for Division

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• The quotient of two positive numbers or two
  negative numbers is a positive number
• -24/-3 8 and 24/3 8

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• The quotient of two numbers when one is positive
  and the other negative is a negative number.
• 24/-3 -8 and -24/3 -8

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PROPERTY

• For all real numbers a and b and nonzero real
  number c,
• (a b)/c a/c b/c
• and
• (a-b)/c a/c b/c

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• 1-7
• Solving Equations in One Variable

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DEFINITION

• Open sentences an equation or inequality
  containing a variable.
• Examples y 1 1 y
• 5x -1 9

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DEFINITION

• Solution any value of the variable that makes
  an open sentence a true statement.
• Examples 2t 1 5
• 3 is a solution or root because
• 23 -1 5 is true

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DEFINITION

• Solution Set the set of all solutions of an
  open sentence. Finding the solution set is
  called solving the sentence.
• Examples y(4 - y) 3
• when y?0,1,2,3
• y ? 1,3

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DEFINITION

• Domain the given set of numbers that a variable
  may represent
• Example
• 5x 1 9
• The domain of x is 1,2,3

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DEFINITION

• Equivalent equations equations having the same
  solution set over a given domain.
• Examples y(4 - y) 3
• when y?0,1,2,3 and
• y2 4y -3 y ? 1,3

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DEFINITION

• Empty set the set with no members and is
  denoted by ?

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DEFINITION

• Identity the solution set is the set of all
  real numbers.

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DEFINITION

• Formula is an equation that states a
  relationship between two or more variables
  usually representing physical or geometric
  quantities.
• Examples d rt
• A lw

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Transformations that Produce Equivalent Equations

• Simplifying either side of an equation.

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• Adding to (or subtracting from) each side of an
  equation the same number or the same expression.

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• Multiplying (or dividing) each side of an
  equation by the same nonzero number.

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• 1-8
• Words into Symbols

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CONSECUTIVE NUMBERS

• Integers n-1, n, n1
• -3, -2, -1, 0, 1, 2, 3,.
• Even Integers n-2, n, n2
• -4,-2, 0, 2, 4,.
• Odd Integers n-2, n, n2
• -5,-3, -1, 1, 3, 5,.

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Addition - Phrases

• The sum of 8 and x
• A number increased by 7
• 5 more than a number

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Addition - Translation

• 8 x
• n 7
• n 5

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Subtraction - Phrases

• The difference between a number and 4
• A number decreased by 8
• 5 less than a number
• 6 minus a number

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Subtraction - Translation

• x - 4
• x- 8
• n 5
• 6 - n

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Multiplication - Phrases

• The product of 4 and a number
• Seven times a number
• One third of a number

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Multiplication - Translation

• 4n
• 7n
• 1/3x

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Division - Phrases

• The quotient of a number and 8
• A number divided by 10

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Division - Translation

• n/8
• n/10

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• 1-9
• Problem Solving with Equations

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Plan for Solving Word Problems

• Read the problem carefully. Decide what numbers
  are asked for and what information is given.
  Making a sketch may be helpful.

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Plan for Solving Word Problems

• Choose a variable and use it with the given facts
  to represent the number(s) described in the
  problem. Labeling your sketch or arranging the
  given information in a chart may help.

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Plan for Solving Word Problems

• Reread the problem. Then write an equation that
  represents relationships among the numbers in the
  problem.

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Plan for Solving Word Problems

1. Solve the equation and find the required numbers.
2. Check your results with the original statement of
   the problem. Give the answer

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EXAMPLES

• Solve using the five-step plan.
• Two numbers have a sum of 44. The larger number
  is 8 more than the smaller. Find the numbers.

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Solution

• n (n 8) 44
• 2n 8 44
• 2n 36
• n 18

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EXAMPLES

• Translate the problem into an equation.
• Marta has twice as much money as Heidi.
• Together they have 36.
• How much money does each have?

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Translation

• Let h Heidis amount
• Then 2h Martas amount
• h 2h 36

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EXAMPLES

• Translate the problem into an equation.
• A wooden rod 60 in. long is sawed into two
  pieces.
• One piece is 4 in. longer than the other.
• What are the lengths of the pieces?

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Translation

• Let x the shorter length
• Then x 4 longer length
• x (x 4) 60

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EXAMPLES

• Translate the problem into an equation.
• The area of a rectangle is 102 cm2.
• The length of the rectangle is 6 cm.
• Find the width of the rectangle?

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Translation

• Let w width of rectangle
• Then 6 length of rectangle
• 6w 102

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EXAMPLES

• Solve using the five-step plan.
• Jason has one and a half times as many books as
  Ramon. Together they have 45 books. How many
  books does each boy have?

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Translation

• Let b number of Ramons books
• Then 1.5b number of Jasons books
• b 1.5b 45

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Solution

• b 1.5b 45
• 2.5b 45
• b 18

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• The End