section 1.2 the real number line. algebraicallyverballynumerical examples graphical example if a is...

Section 1.2

The Real Number Line

Algebraically Verbally Numerical Examples

Graphical Example

If a is a real number, the opposite of a is ____.

Except for zero, the additive inverse of a real number is formed by changing the ______ of the number.

____ is the opposite of 3

____ is the opposite of –3

0 is the opposite of 0

1.2 Lecture Guide: The Real Number Line

Objective 1: Identify additive inverses.

Every real number has an additive inverse. This concept is important when we begin to look carefully at subtraction.

Opposites

−3 30

Opposites or Additive Inverses

The sum of a real number and its additive inverseis _______.

Algebraically Verbally Numerical Examples

Graphical Example

( ) ____

____

a a

a a

3 3 0

3 ____ 0

0 0 0

Opposites or Additive Inverses

−3 30

Opposites

Write the additive inverse of each number:

1. −2 Number:

Additive Inverse: ______

Write the additive inverse of each number:

2. 5Number:

Additive Inverse: ______

Write the additive inverse of each number:

3. 13

Number:

Additive Inverse: ______

Write the additive inverse of each number:

4. Number:

Additive Inverse: ______

0

5.

One number is graphed on each of the following number lines. Graph the additive inverse of each number on the same number line.

0

6.0

−4

5

Double Negative Rule

Algebraically Verbally Numerical Example

For any real number a, .

The opposite of the additive inverse of a is a.

________ a a

7

Simplify each expression.

7. 5.1 5.1 8. 6.4 0

Simplify each expression.

9. 3 10. 3

Objective 2: Evaluate absolute value expressions.

Algebraically Verbally

The absolute value of x is the_________ between 0 and x on the number line.

Numerical Example

Graphical Example

if is nonnegative

if is negative

x xx

x x

2 2

2 2

−2 20

2 units left 2 units right

Absolute Value:

For each number graphed below, determine the distance from this point to the origin, and give the absolute value of this number.

11.

Distance: ______ Absolute value: ______

0−6

For each number graphed below, determine the distance from this point to the origin, and give the absolute value of this number.

12. 0 2

Distance: ______ Absolute value: ______

The absolute value of a nonzero number is always a positive value since distance is never negative.

Evaluate each absolute value expression.

13. 8

1214.

15.

16.

The absolute value of a nonzero number is always a positive value since distance is never negative.

Evaluate each absolute value expression.

8

12

17. If x is positive, the numerical value of the absolute value of x is negative / zero / positive (Circle the best

choice) and x could be represented algebraically by− x / x (Circle the best choice).

18. If x is 0, the absolute value of x is ______.

− x / x (Circle the best choice).

x could be represented algebraically by

If x is negative, the numerical value of the absolute value of x is negative / zero / positive (Circle the bestchoice) and

19.

20.Fill in the blanks to explain why the absolute value of x is defined in two parts. Since distance is never negative, the absolute value of x requires a change in sign for values that are __________________ and does not change the sign for values that are zero or __________________.

Objective 3: Use inequality symbols and interval notation.

Algebraic Notation

Verbal Meaning Graphical Relationship on the Number Line

x equals y x and y are the _______ point.

x is approximately equal to y

x and y are close but are ____________ the same point.

x is not equal to y x and y are ______ points.

x is less than y Point x is to the ____________ of point y.

x is less than or equal to y

Point x is on or to the ____________ of point y.

x y

x y

x y

x y

x y

Equality and Inequality Symbols

Algebraic Notation

Verbal Meaning Graphical Relationship on the Number Line

x is greater than y Point x is to the ____________ of point y.

x is greater than or equal to y

Point x is on or to the ____________ of point y.

Equality and Inequality Symbols

x y

x y

4 ____ 5 3 3_____

4 5

Insert <, =, or > in the blank to make each statement true.

21. 22.

4 _____ 4 5 _____ 5

Insert <, =, or > in the blank to make each statement true.

23. 24.

Insert <, =, or > in the blank to make each statement true.

9 _____ 925.

InequalityNotation

Verbal Meaning Graph IntervalNotation

x is ____________ than a

x is greater than or ____________ to a

x is ____________ than a

x is less than or _______ to a

x is ____________ than a and __________ than b

x a ,a

x a [ , )a

x a

x a ( , ]a

a x b ,a b

Interval Notation

a[

a(

a]

a b

(

( , )a a

)

)

InequalityNotation

Verbal Meaning Graph IntervalNotation

x is ____________ than a and ___________ than or equal to b

x is ____________ than or equal to a and ____________ than b

x is ____________ than or equal to a and ____________ than or equal to b

x is any ____________ number

a x b

a x b

a x b

x ( , )

( , ]a b

[ , )a b

[ , ]a b

Interval Notation

a b

a b

a b

][

)[

](

26. In interval notation a parenthesis means that an endpoint is / is not (Circle the best choice.) included in the interval. A bracket means that an endpoint is / is not (Circle the best choice.) included in the interval.

(

Verbal Description Inequality Notation

Number Line Graph

IntervalNotation

x is greater than three.

x is greater than or equal to – 5 and less than 2.

4x ( ,4]

3(

1 6x -1 6

)

[ 5,2)

27. The table below contains four ways to refer to a set of real numbers. Complete the following table by filling in the missing two columns from each row. It really helps to understand a symbolic notation if you can say the verbal description to yourself.

Objective 4: Mentally estimate square roots and use a calculator to approximate square roots.

Determine without a calculator the exact value to complete each equation.

Estimate the following square roots to the nearest integer and fill in the relationship between the square root and your estimate with either < or >.

Use a calculator or a spreadsheet to approximate the following square roots to the nearest hundredth.

28. Complete the following table of common square roots. To estimate a square root of a number, it is extremely helpful to first think of a perfect square near that number.

1 1

4 2

3

16 4

25 5

6

49

64 8

81

100 10

121

12

169

196

15

11

48

50

125

99

192

170

11

48

50

125

99

192

170

Objective 5: Identify natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

29. Give the definitions of the integers, the whole numbers, and the natural numbers.

Natural Numbers:

Whole Numbers:

Integers:

All real numbers are either rational or irrational.

Rational

Algebraically Numerically Numerical Examples

Verbal Examples

A real number x is rational

if for

integers a and b, with .

In decimal form, a rational number is either a __________ decimal or an infinite repeating decimal.

in decimal form is a terminating decimal.

in decimal form is a repeating decimal.

in decimal form is a repeating decimal.

axb

0b

10.5

2

10.333... 0.3

3

50.151515... 0.15

33

12

13

533

Rational and Irrational Numbers

IrrationalAlgebraically Numerically Numerical

ExamplesVerbal Examples

A real number x is irrational if it cannot be written as

for

integers a and b.

In decimal form, an irrational number is an infinite non-__________ decimal.

cannot be written as a rational fraction – it is an infinite non-repeating decimal. cannot be written as a rational fraction – it is an infinite non-repeating decimal.

This irrational number does exhibit a pattern but it does not terminate and it does not repeat.

Rational and Irrational Numbers

axb

2 1.414213

3.141593

0.1010010001...

2

The following diagram may be helpful to visualize how the subsets of the real numbers are related.

IrrationalNumbers

Rational Numbers

Integers

Whole Numbers

Natural Numbers

The Real Numbers

30. Place a check beneath each column to which each numbers belongs.

Number Natural Whole Integer Rational Irrational Real

0

32

5233

24

16

15

1.234

1.234 1.234234234... means

31. One column in problem 30 has a check mark for each number? Which column? ____________

32. Try evaluating and on a calculator or spreadsheet. What happens?

430

33. Can you express the number 3 as a fraction and in decimal form? If so, provide an example.

34. Is the square root of 4 a rational number?

35. Is the square root of 5 a rational number?