1. Section 1.2
The Real Number Line

2. 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
Opposites or Additive Inverses
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 −3 3

3. Algebraically Verbally Numerical Examples Graphical Example
Opposites or Additive Inverses
Algebraically
Verbally
Numerical Examples
Graphical Example
The sum of a real number and its additive inverse
is _______. −3 3
Opposites

4. Write the additive inverse of each number:1.
Number: −2
Additive Inverse: ______

5. Write the additive inverse of each number:2.
Number:
Additive Inverse: ______

6. Write the additive inverse of each number:3.
Number:
Additive Inverse: ______

7. Write the additive inverse of each number:4.
Number:
Additive Inverse: ______

8. One number is graphed on each of the following number lines
One number is graphed on each of the following number lines. Graph the additive inverse of each number on the same number line. 5. −4 6. 5

9. Double Negative Rule
Algebraically
Verbally
Numerical Example
For any real number a, .
The opposite of the additive inverse of a is a.
________

10. Simplify each expression.7. 8.

11. Simplify each expression.9.
10.

12. Algebraically Numerical Example Graphical Example
Objective 2: Evaluate absolute value expressions.
Absolute Value:
Algebraically
Verbally
The absolute value of x is the_________ between 0 and x on the number line.
Numerical Example
Graphical Example −2 2
2 units left
2 units right

13. Distance: ______ 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. −6
Distance: ______ Absolute value: ______

14. Distance: ______ Absolute value: ______
For each number graphed below, determine the distance from this point to the origin, and give the absolute value of this number.
12. 2
Distance: ______ Absolute value: ______

15. The absolute value of a nonzero number is always a positive value since distance is never negative.
Evaluate each absolute value expression.
13.
14.

16. The absolute value of a nonzero number is always a positive value since distance is never negative.
Evaluate each absolute value expression.
15.
16.

17. 17. If x is positive, the numerical value of the absolute
value of x is negative / zero / positive (Circle the best choice) and
could be represented algebraically by
− x / x (Circle the best choice).
18. If x is 0, the absolute value of x is ______.

18. 19.
If x is negative, the numerical value of the absolute
value of x is negative / zero / positive (Circle the best
choice) and
could be represented algebraically by
− x / x (Circle the best choice).
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 __________________.

19. Objective 3: Use inequality symbols and interval notation.

20. Graphical Relationship on the Number Line
Equality and Inequality Symbols
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
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
Point x is on or to the

21. Graphical Relationship on the Number Line
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

22. Insert <, =, or > in the blank to make each statement true.
21.
22.

23. Insert <, =, or > in the blank to make each statement true.
23.
24.

24. Insert <, =, or > in the blank to make each statement true.
25.

25. Notation Verbal Meaning Graph
Interval Notation
Inequality
Notation
Verbal Meaning
Graph
Interval
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 a ( [ a ) a a ] ( )
a b

26. Inequality Notation Verbal Meaning Graph Interval
Interval Notation
Inequality
Notation
Verbal Meaning
Graph
Interval
x is ____________ than
a and ___________ than
or equal to b
or equal to a and
____________ than b
____________ than or
equal to b
x is any ____________
number ( ]
a b [ )
a b [ ]
a b

27. 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.

28. 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.
Verbal Description
Inequality Notation
Number Line Graph
Interval
Notation
x is greater than
three.
x is greater than or
equal to – 5 and less
than 2. ( 3 ) (

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

30. 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.
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.

31. 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.

32. Algebraically Numerically Numerical Examples Verbal Examples
Rational and Irrational Numbers
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.
repeating

33. Algebraically Numerically Numerical Examples Verbal Examples
Rational and Irrational Numbers
Irrational
Algebraically
Numerically
Numerical Examples
Verbal 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.

34. The following diagram may be helpful to visualize how the subsets of the real numbers are related.
Irrational
Numbers
Rational Numbers
Integers
Whole Numbers
Natural Numbers
The Real Numbers

35. 30. Place a check beneath each column to which each numbers belongs.
Natural
Whole
Integer
Rational
Irrational
Real
means

36. 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?

37. 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?