1. Section 1.2
The Real Number Line

2. 1.2 Lecture Guide: The Real Number Line
Objective: Identify additive inverses.

3. Opposites or Additive Inverses:
Every real number has an additive inverse. This concept is important when we begin to look carefully at subtraction.
Opposites or Additive Inverses:
Algebraically
If a is a real number, the opposite of a is _____.
Verbally
Except for zero, the additive inverse of a real number is formed by changing the ______ of the number.
Numerical Examples
Graphical Example
____ is the opposite of 3 ____ is the opposite of –3 0 is the opposite of 0 −3 3
Opposites

4. Opposites or Additive Inverses:
Algebraically
Verbally
The sum of a real number and its additive inverse is zero.
Numerical Examples

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

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

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

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

9. 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

10. Double Negative Rule:
Algebraically
For any real number a,
Verbally
The opposite of the additive inverse of a is a.
Numerical Examples

11. Simplify each expression.7.

12. Simplify each expression.8.

13. Simplify each expression.9.

14. Simplify each expression.
10.

15. Simplify each expression.
11.

16. Objective: Evaluate absolute value expressions.

17. The absolute value of x is the distance between 0 and x.
Algebraically
Verbally
The absolute value of x is the distance between 0 and x.
Numerical Example
Graphical Example −2 2
2 units left
2 units right

18. 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. −6
Distance: ______ Absolute value: ______

19. 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.
13. 2
Distance: ______ Absolute value: ______

20. The absolute value of a nonzero number is always a positive value since distance is never negative.
Evaluate each absolute value expression and check your results on a calculator.
14.
15.

21. The absolute value of a nonzero number is always a positive value since distance is never negative.
Evaluate each absolute value expression and check your results on a calculator.
16.
17.

22. 18. 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).
19. If x is 0, the absolute value of x is ______.

23. 20. 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).
21. 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 __________________.

24. Objective: Use interval notation
Objective: Use interval notation. Can you list all integers between 1 and 10? Yes, this set of integers is
{2, 3, 4, 5, 6, 7, 8, 9}. Can you list all real numbers between
0 and 1? No, because this set has an infinite number of real
numbers including
. Thus we often use interval
notation as a concise way of referring to a set of real
numbers.

26. 22. 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.
Verbal Description
Inequality Notation
Number Line Graph
Interval Notation
It really helps to understand a symbolic notation if you can say the verbal description to yourself.

27. 22. 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.
Verbal Description
Inequality Notation
x is greater than three.
Number Line Graph
Interval Notation 3 (
It really helps to understand a symbolic notation if you can say the verbal description to yourself.

28. 22. 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.
Verbal Description
Inequality Notation
Number Line Graph
Interval Notation -1 6 ( )
It really helps to understand a symbolic notation if you can say the verbal description to yourself.

29. 18. 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.
Verbal Description
Inequality Notation
x is greater than or equal to −5 and less than 2.
Number Line Graph
Interval Notation
It really helps to understand a symbolic notation if you can say the verbal description to yourself.

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

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

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

33. Insert <, =, or > in the blank to make each statement true.
26.

34. Insert <, =, or > in the blank to make each statement true.
27.

35. Objective: Estimate and approximate square roots.
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.

36. Estimate the following square roots to the nearest integer and fill in the relationship between the number and your estimate with either < or >.
Use your calculator to approximate the following square roots to the nearest hundredth.

37. Estimate the following square roots to the nearest integer and fill in the relationship between the number and your estimate with either < or >.
Use your calculator to approximate the following square roots to the nearest hundredth.

38. Estimate the following square roots to the nearest integer and fill in the relationship between the number and your estimate with either < or >.
Use your calculator to approximate the following square roots to the nearest hundredth.

39. 29. Use a calculator to complete the following table
29. Use a calculator to complete the following table. (Hint: See Calculator Perspective )

40. Objective: Identify natural numbers, whole numbers, integers, rational numbers, and irrational numbers. All real numbers are either rational or irrational.

41. Rational and Irrational Numbers
Algebraically
A real number x is rational if
for integers a and b, with
Numerically
In decimal form, a rational number is either a terminating decimal or an infinite repeating decimal.
Numerical Examples
Verbal Examples
in decimal form is a terminating decimal.
in decimal form is a repeating decimal.
in decimal form is a repeating decimal.

42. Irrational
Algebraically
A real number x is irrational if it cannot be written as
for integers a and b.
Numerically
In decimal form, an irrational number is an infinite non-repeating decimal.
Numerical Examples
Verbal Examples
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.

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

44. 31. Give the definitions of the integers, the whole numbers, and the natural numbers.

45. 32. Is the square root of 4 a rational number?

46. 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

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

48. 35. Try evaluating
and
on your calculator.
What happens?