Basic Concepts 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 BA

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

set of counting 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 .

Each point on a number line is paired with exactly one real number, called the coordinate of the point. 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 - 3 24 ÷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 - 3 24 ÷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

≥ - greater than or equal to INEQUALITY SYMBOL One of the symbols < - less than > greater than ≠ - does not equal ≤ - less than or equal to ≥ - greater than or equal to

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

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

the result of subtracting one number from another DIFFERENCE the result of subtracting one number from another 8 – 6 = 2 10 - 2 = 8 difference

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

the result of dividing one number by another QUOTIENT the result of dividing one number by another 35 ÷ 7 = 5 10 ÷ 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 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}

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

If the domain of x is {0,1,2,3}, we write 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 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.

Perform multiplications and divisions in order from left to right. and ORDER OF OPERATIONS Grouping symbols Simplify powers Perform multiplications and divisions in order from left to right. and

ORDER OF OPERATIONS Perform additions and subtractions in order from left to right 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 = a Symmetric Property - If a = b, then b = a Transitive 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 unique 7 + 5 = 12 7 x 5 = 35

COMMUTATIVE PROPERTIES a + b = b + a ab = ba 2 + 6 = 6 + 2 2 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)

There are unique real numbers 0 and 1 (1≠0) such that: 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 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

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 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

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

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

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

The product of two positive numbers or two negative numbers is a positive number.

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

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

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

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

The quotient of two numbers when one is positive and the other negative is a negative number.

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

DEFINITION Open sentences – an equation or inequality containing a variable. Examples: y + 1= 1 + y 5x -1 = 9

3 is a solution or root because 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 2·3 -1= 5 is true

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}

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) = 3 when y{0,1,2,3} and y2 – 4y = -3 y  {1,3}

DEFINITION Empty set – the set with no members and is denoted by 

DEFINITION Identity – the solution set is the set of all real numbers.

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

Transformations that Produce Equivalent Equations Simplifying either side of an equation.

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

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

1-8 Words into Symbols

CONSECUTIVE NUMBERS Integers – {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,….}

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

The difference between a number and 4 A number decreased by 8 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 1/3x

The quotient of a number and 8 A number divided by 10 Division - Phrases The quotient of a number and 8 A number divided by 10

Division - Translation

1-9 Problem Solving with Equations

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.

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.

Plan for Solving Word Problems Reread the problem. Then write an equation that represents relationships among the numbers in the problem.

Plan for Solving Word Problems Solve the equation and find the required numbers. Check your results with the original statement of the problem. Give the answer

Solve using the five-step plan. 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. Marta has twice as much money as Heidi. Together they have $36. How much money does each have?

Let h = Heidi’s amount Then 2h = Marta’s amount h + 2h = 36 Translation Let h = Heidi’s amount Then 2h = Marta’s amount h + 2h = 36

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?

Let x = the shorter length Then x + 4 = longer length x + (x + 4) = 60 Translation Let x = the shorter length Then x + 4 = longer length x + (x + 4) = 60

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?

Let w = width of rectangle Then 6 = length of rectangle 6w = 102 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?

Let b = number of Ramon’s books Then 1.5b = number of Jason’s books 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