Mt1nt_02
tran wk bk
7/22/03
9:22 AM
2.1
Page 21
The Real Number Line
Goals p Graph and compare real numbers using a number line.
p Find the opposite and the absolute value of a number.
VOCABULARY
Real numbers Whole numbers, decimals, and fractions are
examples of real numbers.
Real number line A real number line is a horizontal line that
pictures real numbers as points.
Negative numbers Negative numbers are any of the numbers
less than zero.
Positive numbers Positive numbers are any of the numbers
greater than zero.
Integers Any of the numbers …, ⫺3, ⫺2, ⫺1, 0, 1, 2, 3, … are
integers.
Graph The point that corresponds to a number is the graph
of the number.
Plotting Drawing the point is called graphing the number or
plotting the point.
Opposites Two points on a number line that are the same
distance from the origin but are on opposite sides of it are
opposites.
Absolute value The absolute value of a real number is the
distance between the origin and the point representing the
real number.
Velocity Velocity is the speed and direction in which an
object is traveling (up is positive and down is negative).
Counterexample A counterexample is an example used to
show that a given statement is false.
Lesson 2.1 • Algebra 1 Notetaking Guide
21
Mt1nt_02
tran wk bk
7/22/03
9:22 AM
Page 22
Example 1
Comparing Real Numbers
Graph ⫺3 and ⫺7 on a number line. Then write two inequalities
that compare the two numbers.
Solution
⫺7
⫺8
⫺3
⫺7
⫺6
⫺5
⫺4
⫺3
⫺2
⫺1
0
1
2
On the graph, ⫺7 is to the left of ⫺3, so ⫺7 is less than ⫺3.
On the graph, ⫺3 is to the right of ⫺7, so ⫺3 is greater than
⫺7.
Answer ⫺7 < ⫺3 and ⫺3 > ⫺7.
Example 2
Ordering Real Numbers
4
3
Write the following numbers in increasing order: ⫺ᎏᎏ, 1.8, ᎏᎏ, ⫺1.
5
2
Solution Graph the numbers on a number line.
4
3
2
⫺5
⫺1
⫺3
⫺2
⫺1
1.8
0
1
2
3
Answer From the graph, you can see that the order is
4 3
⫺1, ⫺ᎏᎏ, ᎏᎏ, 1.8
5 2
.
Example 3
Finding the Opposite of a Number
Find the opposite of ⫺4.
Solution
4
⫺5
⫺4
⫺3
⫺2
4
⫺1
0
1
2
3
4
5
Answer The opposite of ⫺4 is 4 because ⫺4 and 4 are on
opposite sides of the origin and are both 4 units from the origin.
22
Algebra 1 Notetaking Guide • Chapter 2
Mt1nt_02
tran wk bk
7/22/03
9:22 AM
Page 23
THE ABSOLUTE VALUE OF A NUMBER
p If a is a positive number, then a a .
Example: 10 10
p If a is zero, then a 0 .
Example: 0 0
p If a is a negative number, then a a .
Example: 10 10
Example 4
Solving an Absolute Value Equation
Use mental math to solve the equation.
a. x 3
b. x 2
Solution
a. The numbers 3 and 3 are 3 units from the origin, so there
are two solutions: 3 and 3 .
b. Because distance is never negative, the absolute value of a
number is never negative, so there is no solution .
Example 5
Using a Counterexample
Decide whether the statement is true or false. If it is false, give a
counterexample.
a. The expression a is sometimes greater than a.
b. The expression a is always less than a.
Solution
a. True. When a < 0, a is greater than a.
b. False. Counterexample: If a 3, then a 3, which
is not less than 3.
Checkpoint Complete the following exercise.
1. Compare 4 and 4 using <, >, or .
|4| < 4
Lesson 2.1 • Algebra 1 Notetaking Guide
23