INTERMEDIATE ALGEBRA ninth edition by Lial | Hornsby | McGinnis

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INTERMEDIATE ALGEBRA ninth edition by Lial | Hornsby | McGinnis. © 2005 Presentation by Vernon McBride. INTERMEDIATE ALGEBRA ninth edition by Lial | Hornsby | McGinnis. Chapter 1 Review of the Real Number System. 1.1 Basic Concepts. Objective 1 Write sets using set notation

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### INTERMEDIATE ALGEBRAninth edition by Lial | Hornsby | McGinnis

by Vernon McBride

### INTERMEDIATE ALGEBRAninth edition by Lial | Hornsby | McGinnis

Chapter1

Review of the

Real Number System

1.1 Basic Concepts

Objective 1 Write sets using set notation

A set is a collection of objects.

• The objects are called elements or members of the set.
• Set braces, { }, are used to enclose the elements.
• If we can count the number of elements in a set such as {1,2,3}, it is a finite set.

The numbers we use to count things are the set of natural numbers, N= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}

• The three dots, ellipses, show that the same pattern continues in the same pattern indefinitely.
• Each natural number other than 1 is either a prime number or a composite number.
• A prime number is a number greater than one that is evenly divisible only by itself or 1.
• A natural number other than 1 that is not a prime number is a composite number.

The whole numbers include 0 and the natural numbers, W= {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …}

A set containing no elements, such as the set of whole numbers less than zero, is called the empty set, or null set, which is written  or { }.

1.1 Basic Concepts

Objective 1Write sets using set notation

Consider these sentences:

• To do well in math you must do the homework and it takes a lot of time.
• I earned a college degree, and it allowed me to get a better paying job.

In the first sentence “ it ” means homework; in the second sentence “it ” means college degree.

Similarly, in mathematics, a letter of the alphabet can be used to stand for some number.

• A letter used in this way is called a variable.

In algebra, letters called variables, are often used to represent numbers or to define sets of numbers.

For example, in set-builder notation,

{x | x is a natural number between 4 and 18}

(read “the set of all elements x such that x is a natural number between 4 and 18”) defines the set

{5, 6, 7, …, 17}, where we can say

7  {5, 6, 7, …, 17} ( read “ is an element of”).

1.1 Basic Concepts

Objective 2 Use number lines.

A good way to get a picture of a set of numbers is to use a real number line.

To construct a number line,

• Choose any point on a horizontal line and label it 0.
• Choose a point to the right of 0 and label it 1.
• The distance from 0 to 1 establishes a scale that can be used to locate more points, with positive numbers to the right of 0 and negative numbers to the left of 0.
• The number 0 is neither positive nor negative.
• The set of numbers including positive and negative counting numbers and zero are called the set of integers:
• The negative integers are {…,–5,–4,–3,–2,–1}.
• The positive integers are {1, 2, 3, 4, 5, …}.

Z= {…, –5 , –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, …}

• Each number on a number line is called the coordinate of the point that it labels while the point is the graph of the number.
• A rational number can be expressed as the quotient of two integers, with the denominator not 0.
• Rational numbers can also be written in decimal form, either as terminating; ¾=.75, -⅛ = -.125, or 1¼= 2.75
• or as repeating decimals; ⅓ = .33333… or ⅙ = .1666….
• Decimal numbers that neither terminate nor repeat are not rational, and thus are called irrational numbers.
• The square root of any number that is not a perfect square is an irrational number; as well as numbers represented by familiar symbols;  or e.

Real Numbers ( R )

(an aggregate set of subsets)

Rational Numbers ( Q )

Irrational Numbers ( I)

can be expressed as an Integer divided by an integer; either terminating or repeating in decimal form.

neither repeat nor terminate; cannot be expressed as an integer divided by an integer.

.

Whole Numbers ( W )

.

Integers ( Z )

zero with the set of Natural Numbers

Negative Integers

opposites of the counting numbers

Natural Numbers ( N )

the counting numbers.

add the opposites of the counting numbers to the set of whole numbers

1.1 Basic Concepts

Objective 3 Know the common sets of numbers.

1.1 Basic Concepts

For each positive number, there is a negative number on the opposite side of 0 that lies the same distance from 0.

• These pairs of numbers are called the additive inverses, negatives, or opposites of each other.
• The sum of a number and its additive inverse is always 0.
• The number 0 is its own additive inverse.
• Numbers written with positive or negative signs are called signed numbers.
• A positive number can be called a signed number even though the positive sign is usually left off.

For any real numbera, the number – a is the additive inverse of a.

For any real number a, – (– a) = a.

1.1 Basic Concepts

Objective 5 Use Absolute Value.

Geometrically, the absolute value of a number a, written |a|, is the distance on the number line from 0 to a.

• If a is a negative number, then – a, the additive inverse or opposite ofa, is a positive number, so |a| is positive.

The projected annual rates of employment change (in percent) in some of the fastest growing and most rapidly declining industries from 1994 through 2005 are shown in the table.

• What industry in the list is expected to see the greatest change? the least change?
• We want the greatest change, without regard to whether the change is an increase or a decrease.

Absolute Value

1.1 Basic Concepts

Objective 6 Use inequality symbols.

• A statement comparing expressions that represent two quantities as equal is an equation; 2 + 3 = 5.
• A statement that two quantities are equal is an inequality; 3 ≠ 7 (read “3 is not equal to 7”).
• When two numbers are not equal, one must be less than the other.
• the symbol < means “is less than”; 8< 9, –2< 5, and – 9<– 4
• the symbol > means “is greater than”; 11>7, 2 > –3, and ¾>0

Notice that in each case, the symbol “points toward the smaller number.

• The smaller of two numbers is always to the left of the other on a number line.

Inequalities on a Number Line

On a number line,

a < b if ais to the left of b; a > b if ais to the right of b;

1.1 Basic Concepts

Objective 7Graph sets of real numbers.

• In set-builder notation, the set of integers greater than – 2 is written

{x | x > – 2, x  integers }

and is read “the set of all x such that x is greater than – 2 and x is an element of the integers.”

This is an infinite set. It is impossible to list all the elements of the set.

• The set of real numbers less than or equal to 4 is written

{x | x ≤ 4, x  real numbers }

and is read “the set of all x such that x is less than or equal to 4 and x is an element of the real numbers.”

• The set of all real numbers greater than – 2 is an example of an interval on the number line.
• To write intervals, we use interval notation:

Set-builder Interval notation

{x | x > – 2} = (– 2, )

• It is common to graph two intervals or sets of numbers that are between two given numbers which are expressed as compound inequalities.

Open

Interval Notation for Unbounded Intervals

(

a

Set-builder notation

Interval notation

Type

Graph

Open

)

a

Half Open

[

a

Half Open

]

a

Real Numbers

Open

Compound Inequality

] (

a

b

Interval Notation for Bounded Intervals(Compound Inequalities)

Set-builder notation

Interval notation

Type

Graph

Open

( )

a

b

Closed

[]

a

b

[)

Half Open

a

b

Half Open

(]

a

b

1.1 Basic Concepts

Objective 7Graph sets of real numbers.

• A compound inequality is two or more inequalities connected by the word and or or.
1.2 Operations on Real Numbers

Recall that the answer to an addition problem is called the sum.

Finding the sum:

• – 11 + (– 9) First, find the absolute values.

|– 11| = 11 and |– 9| = 9 Same sign, add absolute values.

– 11 + (– 9) = – (11 + 9) = – (20) = – 20 Answer same sign.

• Write each fraction with LCD.

Different signs, subtract abs values.

Larger abs value is negative.

Like signs To add two numbers with the same sign, add their absolute values.

The sign of the answer (either + or – ) is the same as the sign of the two numbers.

Unlike signsTo add two numbers with the different signs, subtract the smaller absolute value from the larger.

The sign of the answer is the same as the sign of the number with the larger absolute value.

1.2 Operations on Real Numbers

Objective 2Subtract real numbers.

Recall that the answer to a subtraction problem is called the difference.

(Subtraction is just like adding the inverse or opposite of the number.)

Finding the difference:

• 5 – 9 = 5 + (– 9) To subtract just add the inverse.

= – (9 – 5) Different signs, subtract abs values.

= – 4 Larger absolute value is negative.

• – 9 –(– 3) = – 9 +[–(– 3)] Add the opposite of the inverse.

= – 9 + 3 The opposite of the inverse is positive.

= – (9 – 3) Different signs, subtract abs. values.

= – 6 Larger absolute value is negative.

• Add the opposite of the inverse.

• When working problems that involves both addition and subtraction, add and subtract from left to right.
• Work inside brackets or parenthesis first.

Subtraction

For all real numbers a and b.

a – b = a + ( – b).

That is, change the sign of the second number and add.

1.2 Operations on Real Numbers

Objective 3 Find the distance between two points on the number line.

To find the distance between two points on the number line we subtract the coordinates of the points.

• Since distance is always positive (or 0), we must be careful to subtract in such a way that the answer is positive (or 0).
• Or, to avoid this problem altogether, we can find the absolute value of the difference.

Objective 4 Multiply real numbers.

The answer to a multiplication problem is called the product.

Distance

The distance between two points on a number line is the absolute value of the difference between the numbers.

Multiplying Real Numbers

Like signs The product of two numbers with the same sign is positive.

Unlike signsThe product of two numbers with different signs is negative.

1.2 Operations on Real Numbers

Objective 5 Divide Real Numbers.

• The result of dividing one number by another number is called the quotient.
• The quotient of two real numbers a  b (b ≠ 0) is the real number qsuch that q  b = a. That is,

a  b = q only if q  b = a.

There is no number whose product with 0 gives 5.

because any number multiplied by 0 gives 0.

When dividing we always want a unique quotient, and therefore division by 0 is undefined.

• Thus, dividing by b is the same as multiplying by

If b≠ 0, then is the reciprocal (or multiplicative inverse) of b.

• When multiplied, reciprocals have a product of 1.
• There is no reciprocal for 0 because there is no number that can be multiplied by 0 to give a product of 1.
1.2 Operations on Real Numbers

Objective 5 Divide Real Numbers.

• Since division is defined as multiplication by the reciprocal, the rules for signs of quotients are the same as those for signs of products.
• Every fraction has three signs: The sign of the numerator, the sign of the denominator, and the sign of the fraction itself.
• Changing any two of these three signs does not change the value of the fraction.

Division

For all real numbers a and b (where b ≠ 0),

That is, multiply the first number by the reciprocal of the second number.

Dividing Real Numbers

Like signs The quotient of two nonzero real numbers with the same sign is positive.

Unlike signsThe quotient of two nonzero real numbers with different signs is negative.

Equivalent Forms of a Fraction (Assuming y ≠ 0)

1.3 Exponents, Roots, and Order of Operations

Objective 1 Use exponents.

• Two or more numbers whose product is a third number are factors of that third number.
• In algebra, we use exponents as a way of writing products of repeated factors.
• The term squared comes from the figure of a square, which has the same measure for both length and width.
• Similarly, the term cubed comes from the figure of a cube, whose length, width, and height have the same measure.

Exponential Expression

If a is a real number and n is a natural number, then

where n is the exponent, a is the base, and a nis anexponential expression.

Exponents are also called powers.

The product of an odd number of negative factors is negative.

The product of an even number of negative factors is positive.

1.3 Exponents, Roots, and Order of Operations

Objective 2 Find square roots.

• The opposite (inverse) of squaring a number is called taking its square root.
• We write the positive or principal square root of a number with the symbol , called a radical sign.
• Since the square of any nonzero real number is positive, the square root of a negative number is not a real number.

Objective 3 Use the order of operations.

• When an expression involves more than one operation symbol, we use the following order of operations.
• Order of Operations
• Work separately above and below any fraction bar.
• If grouping symbols such as parentheses ( ), square brackets [ ], or absolute value bars | | are present, start with the innermost set and work outward.
• Evaluate all powers, roots, and absolute values.
• Do any multiplications or divisions in order, working from left to right.
• Do any additions or subtractions in order, working from left to right.
1.3 Exponents, Roots, and Order of Operations

Objective 4 Evaluate algebraic expressions for given values of variables.

• Any collection of numbers, variables, operation symbols, and grouping symbols, such as;

5ab, 3n – m, and 5(x2 – 2y),

is called an algebraic expression.

• Algebraic expressions have different numerical values for different values of the variables.
• We can evaluate such expressions by substituting given values for the variables.

3

4

2

2

1.4 Properties of Real Numbers

Objective 1 Use the distributive property.

• The study of any object is simplified when we know the properties of the object.
• The basic properties of real numbers reflect results that occur consistently in work with numbers.
• In algebra, we generalize these properties to apply to expressions with variables.

Notice that 2(3 + 4) = 2  7 = 14

and 2  3 + 2  4 = 6 + 8 = 14

So that 2(3 + 4) = 2  3 + 2 4

These arithmetic examples are generalized to all real numbers as the distributive property of multiplication with respect to addition, or simply the distributive property.

Distributive Property

For any real numbers a, b, and c.

a(b + c) = ab + ac and (b + c)a = ba + ca

1.4 Properties of Real Numbers

Objective 2 Use the inverse properties.

• The additive inverse of a number a is – aand the sum of a number and its additive inverse is 0.
• The number 0 is its own additive inverse.
• Two numbers with a product of 1 are reciprocals and another name for reciprocal is multiplicative inverse.

Inverse Properties

For any real number a, there is a single real number – a,

such that

a + (– a) = 0and– a + a = 0.

The inverse “undoes” addition with the result 0.

For any nonzero real number a, there is a single real number such that

The inverse “undoes” multiplication with the result 1.

1.4 Properties of Real Numbers

Objective 3 Use the identity properties.

• Zero is the only number that can be added to any number to get that number, so adding 0 leaves the identity of the number unchanged. 0 is called the identity element for addition or the additive identity.
• Multiplying by 1 leaves the identity of any number unchanged, so 1 is the identity element for multiplication or the multiplicative identity.

Identity Properties

For any real number a

a + 0 =0 + a = a.

The answer is “identical” to a.

Also,

a  1 = 1  a = a.

The answer is “identical” to a.

1.4 Properties of Real Numbers

Objective 4 Use the commutative and associative properties.

• A term is a number or the product of a number and one or more variables.
• Terms with exactly the same variables raised to exactly the same powers are called like terms (or similar terms).
• The numerical factor in a term is called the numerical coefficient, or just the coefficient.
• To simplify expressions we combine like terms.
• The commutative properties are used to change the order of the terms or factors.
• The associative properties are used to regroup the terms or factors of an expression.

Commutative and Associative Properties

For any real number a, b, and c,

Commutative properties

Reverse the order of two terms or factors.

Also,

Associative properties

Shift parentheses among three terms or factors; order stays the same.

1.4 Properties of Real Numbers

Objective 5 Use the multiplication property of 0.

• The additive identity property gives a special property of 0, namely that a + 0 = a for any real number a.
• The multiplication property of 0 gives a special property of 0 that involves multiplication:

The product of any real number and 0 is 0.

Multiplication Property of 0

For any real number a

a  0 = 0 and 0  a = 0

Chapter 1 - Homework Assignments

Exercises and Applying Concepts

1.1 Basic Concepts

Do problems:1, 6,4, 8, 9, 11, 14, 16, 25, 30, 35, 40, 45, 50, 55, 81, 86, 91, 96, 100, 101, 106, 111

1.2 Operations on Real Numbers

Do problems:10, 11, 13, 16, 20, 23, 25, 30, 32, 34, 39, 42, 46, 50, 61, 68, 77, 83, 86, 94

1.3 Exponents, Roots, and Order of Operations

Do problems:2, 11, 18, 22, 27, 30, 33, 38, 41, 46, 48, 49, 57, 66, 75, 81, 91

1.4 Properties of Real Numbers

Do problems:13, 16, 24, 28, 32, 36, 40, 41, 48, 50, 52