Chapter 1 Basic Concepts of Algebra LANGUAGE OF ALGEBRA.
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Transcript of Chapter 1 Basic Concepts of Algebra LANGUAGE OF ALGEBRA.
Chapter 1
Basic Concepts of Algebra
LANGUAGE OF ALGEBRA
SET– a collection or group of, things, objects,
numbers, etc.
INFINITE SET – a set whose members cannot
be counted.
If A= {1, 2, 3, 4, 5,…} then A is infinite
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
SUBSET – all members of a set are members of
another set
If A= {e, f, g, h, i, j} and B = {e, i} ,
then BA
EMPTY SET or NULL SET – a set having no
elements.
A= { } or B = { } are empty sets or null sets written as
1-1
Real Numbers and Their Graphs
Real Numbers
NATURAL NUMBERS - set of counting numbers
{1, 2, 3, 4, 5, 6, 7, 8…}
WHOLE NUMBERS - set of counting numbers plus
zero
{0, 1, 2, 3, 4, 5, 6, 7, 8…}
INTEGERS - set of the whole numbers plus their
opposites
{…, -3, -2, -1, 0, 1, 2, 3, …}
RATIONAL NUMBERS - numbers that can be
expressed as a ratio of two integers a and b and
includes fractions, repeating decimals, and terminating
decimals
EXAMPLES OF RATIONAL NUMBERS
½, ¾, ¼, - ½, -¾, -¼, .05
.76, .333…, .666…, etc.
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
EXAMPLES OF IRRATIONAL NUMBERS
, 6, -29, 8.11211121114…,
etc.
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
1-2
Simplifying Expressions
Definitions
NUMERICAL EXPRESSION or NUMERAL
a symbol or group of symbols used to represent a number
3 x 4 5 + 5 +2 15 - 324 ÷2 12 2 x 6
VALUE of a Numerical Expression
The number represented by the expression
Twelve is the value of 3 x 4 5 + 5 +2 15 - 324 ÷2 12 2 x 6
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.
EXAMPLES OF EQUATIONS
-6 + 10 = 6 – 2 or 4x + 3 = 19
INEQUALITY SYMBOL One of the symbols
< - less than > greater than
≠ - does not equal≤ - less than or equal to
≥ - greater than or equal to
INEQUALITYa sentence formed by placing an inequality symbol between
two expressions, called the sides of the inequality
-3 > -5-3 < - 0.3
SUMthe result of adding numbers, called the terms of the sum
6 + 15 = 2110 + 2 = 12
terms sum
DIFFERENCEthe result of subtracting one
number from another8 – 6 = 210 - 2 = 8
difference
PRODUCTthe result of multiplying
numbers, called the factors of the product6 x 15 = 8010 · 2 = 20
factors product
QUOTIENTthe result of dividing one
number by another35 ÷ 7 = 510 ÷ 2 = 5
quotient
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.
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
GROUPING SYMBOLSPairs 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}
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
EXAMPLES OF VARIABLES
If the domain of x is {0,1,2,3}, we write
x {0,1,2,3}
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,…
Algebraic Expression – a numerical expression; a
variable; or a sum, difference, product, or
quotient that contains one or more variables
EXAMPLES OF ALGEBRAIC
EXPRESSIONS 24 + 3 + x y2 – 2y + 6
a + b 2c2d – 4 c d
SUBSTITUTION PRINCIPLE
An expression may be replaced by another
expression that has the same value.
ORDER OF OPERATIONS1. Grouping symbols
2. Simplify powers
3. Perform multiplications and divisions in order from left to right. and
ORDER OF OPERATIONS4. Perform additions and
subtractions in order from left to right
5. Simplify the expression within each grouping symbol, working outward from the innermost grouping
DEFINITION of ABSOLUTE VALUE
For each real number a,
l a l = a if a >0
0 if a = 0
- a if a < 0
1-3
Basic Properties of Real Numbers
Properties of Equality
Reflexive Property - a = aSymmetric Property - If a =
b, then b = aTransitive Property - If a = b,
and b = c, then a = c
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
Properties of Real Numbers
CLOSURE PROPERTIES
a + b and ab are unique7 + 5 = 127 x 5 = 35
COMMUTATIVE PROPERTIES
a + b = b + a
ab = ba
2 + 6 = 6 + 22 x 6 = 6 x 2
ASSOCIATIVE PROPERTIES
(a + b) + c = a + (b
+c)(ab)c = a(bc)(5 + 15) + 20 = 5 + (15
+20)(5·15)20 = 5(15·20)
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
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
INVERSE PROPERTIESPROPERTY 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
DISTRIBUTIVE PROPERTY
a(b + c) = ab + ac(b +c)a = ba + ca
5(12 + 3) = 5•12 + 5 •3 = 75
(12 + 3)5 = 12• 5 + 3 • 5 = 75
1-4
Sums and Differences
Rules for Addition
For real numbers a and b1. If a and b are negative
numbers, then a + b is negative and a + b = -(lal + lbl) -5 + (-9) = - (l-5l + l-9l) = -14
For real numbers a and b2. 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
For real numbers a and b3. 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
DEFINITION of SUBTRACTION
For all real number a and b,
a – b = a + (-b)
To subtract any real number, add its opposite
DISTRIBUTIVE PROPERTY
For all real number a ,b, and c
a(b - c)= ab – ac
and
(b – c)a = ba - ca
1-5
Products
MULTIPLICATIVE PROPERTY OF 0
For every real number a,
a · 0 = 0 and 0 · a = 0
MULTIPLICATIVE PROPERTY OF -1
For every real number a,a(-1) = -a and (-1)a = -a
Rules for Multiplication
1. The product of two positive numbers or two negative numbers is a positive number.(5)(9) = 45 or (-5)(-9) = 45
2. The product of a positive number and a negative number is a negative number.(-5)(9) = -45 or (5)(-9) = -45
3. The absolute value of the product of two or more numbers is the product of their absolute valuesl(-5)(9)l = l-5l l9l = 45
PROPERTY of the OPPOSITE of a
PRODUCT For all real number a and
b,-ab = (-a)b
and-ab = a(-b)
PROPERTY of the OPPOSITE of a SUM
For all real number a and
b,-(a + b) = (-a) + (-b)
1-6
Quotients
DEFINITION OF DIVISION
The quotient a divided by b is written a/b or a÷b. For every real number a and nonzero real number b,
a/b = a·1/b, or a÷b = a·1/b
DEFINITION OF DIVISION
To divide by any nonzero number, multiply by its reciprocal. Since 0 has no reciprocal, division by 0 is not defined.
Rules for Division
1. The quotient of two positive numbers or two negative numbers is a positive number-24/-3 = 8 and 24/3 = 8
2. The quotient of two numbers when one is positive and the other negative is a negative number.
24/-3 = -8 and -24/3 = -8
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
1-7 Solving Equations in One
Variable
DEFINITIONOpen sentences – an equation or inequality containing a variable.
Examples: y + 1= 1 + y
5x -1 = 9
DEFINITION
Solution – any value of the variable that makes an open sentence a true statement.
Examples: 2t – 1 = 53 is a solution or root because
2·3 -1= 5 is true
DEFINITIONSolution Set – the set of all
solutions of an open sentence. Finding the solution set is called solving the sentence.
Examples: y(4 - y) = 3when y{0,1,2,3}
y {1,3}
DEFINITION
Domain – the given set of numbers that a variable may represent
Example:
5x – 1 = 9
The domain of x is {1,2,3}
DEFINITION
Equivalent equations – equations having the same solution set over a given domain.
Examples: y(4 - y) = 3when y{0,1,2,3} and
y2 – 4y = -3 y {1,3}
DEFINITION
Empty set – the set with no members and is denoted by
DEFINITIONIdentity – the solution set is
the set of all real numbers.
DEFINITIONFormula – is an equation that
states a relationship between two or more variables usually representing physical or geometric quantities.
Examples: d = rtA = lw
Transformations that Produce Equivalent
Equations1. Simplifying either side of an
equation.
2. Adding to (or subtracting from) each side of an equation the same number or the same expression.
3. Multiplying (or dividing) each side of an equation by the same nonzero number.
1-8
Words into Symbols
CONSECUTIVE NUMBERSIntegers – {n-1, n, n+1}
{… -3, -2, -1, 0, 1, 2, 3,….}
Even Integers – {n-2, n, n+2}{…-4,-2, 0, 2, 4,….}
Odd Integers – {n-2, n, n+2}{…-5,-3, -1, 1, 3, 5,….}
Addition - Phrases
• The sum of 8 and x• A number increased by 7• 5 more than a number
Addition - Translation
• 8 + x• n + 7• n + 5
Subtraction - Phrases
• The difference between a number and 4
• A number decreased by 8• 5 less than a number• 6 minus a number
Subtraction - Translation
• x - 4• x- 8• n – 5• 6 - n
Multiplication - Phrases
• The product of 4 and a number
• Seven times a number• One third of a number
Multiplication - Translation
• 4n• 7n• 1/3x
Division - Phrases
• The quotient of a number and 8
• A number divided by 10
Division - Translation
• n/8• n/10
1-9 Problem Solving with
Equations
Plan for Solving Word Problems
1. Read the problem carefully. Decide what numbers are asked for and what information is given. Making a sketch may be helpful.
Plan for Solving Word Problems2. 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.
Plan for Solving Word Problems3. Reread the problem. Then
write an equation that represents relationships among the numbers in the problem.
Plan for Solving Word Problems4. Solve the equation and find the
required numbers.
5. Check your results with the original statement of the problem. Give the answer
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.
Solution
n + (n + 8) = 44
2n + 8 = 44
2n = 36
n = 18
EXAMPLES
Translate the problem into an equation.
(1) Marta has twice as much money as Heidi.
(2) Together they have $36.How much money does each have?
Translation
Let h = Heidi’s amount
Then 2h = Marta’s amount
h + 2h = 36
EXAMPLES
Translate the problem into an equation.
(1) A wooden rod 60 in. long is sawed into two pieces.
(2) One piece is 4 in. longer than the other.
What are the lengths of the pieces?
Translation
Let x = the shorter length
Then x + 4 = longer length
x + (x + 4) = 60
EXAMPLES
Translate the problem into an equation.
(1) The area of a rectangle is 102 cm2.
(2) The length of the rectangle is 6 cm.
Find the width of the rectangle?
Translation
Let w = width of rectangle
Then 6 = length of rectangle
6w = 102
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?
Translation
Let b = number of Ramon’s books
Then 1.5b = number of Jason’s books
b + 1.5b = 45
Solution
b + 1.5b = 45
2.5b = 45
b = 18
The End