the number line

1. The Number Line

2. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.

3. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
0
the origin

4. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East)
20 1 3
+
the origin

5. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
the origin

6. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
the origin
2½

7. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
2/3 2½
the origin

8. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..–π  –3.14..
the origin

9. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a number is called
the number line.
–π  –3.14..
the origin

10. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a number is called
the number line. Given two numbers and their positions on the
number line, we define the number R to the right to be greater
than the number L to the left and we write that “L < R”.
–π  –3.14..
the origin
+– L R

11. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a number is called
the number line. Given two numbers and their positions on the
number line, we define the number R to the right to be greater
than the number L to the left and we write that “L < R”.
–π  –3.14..
the origin
+– L R<

12. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a number is called
the number line. Given two numbers and their positions on the
number line, we define the number R to the right to be greater
than the number L to the left and we write that “L < R”.
–π  –3.14..
the origin
+– –1–2
<For example,
–2 is to the left of –1,
so written in the natural–form “–2 < –1”.
0
L R<

13. The Number Line
Just like assigning address to houses on a street we assign
addresses to points on a line.
We assign 0 to the “center” of the line, and we call it the origin.
We assign the directions with signs, positive numbers to the
right (East) and negative numbers to the left (West).
-2 20 1 3
+
-1-3
–
2/3 2½ π  3.14..
This line with each position addressed by a number is called
the number line. Given two numbers and their positions on the
number line, we define the number R to the right to be greater
than the number L to the left and we write that “L < R”.
–π  –3.14..
the origin
+– –1–2
<For example,
–2 is to the left of –1,
so written in the natural–form “–2 < –1”. This may be written
less preferably in the reversed direction as –1 > –2.
0
L R<

14. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
The Number Line

15. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
The Number Line

16. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x".
The Number Line

17. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a.
The Number Line

18. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a. In picture,
+–
a
open dot
The Number Line

19. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a. In picture,
+–
a
open dot
a < x
The Number Line

20. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a. In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x.
a < x
The Number Line

21. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a. In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x
The Number Line

22. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a. In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.
The Number Line

23. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a. In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.
+–
a b
The Number Line

24. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a. In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b.
+–
a a < x < b b
The Number Line

25. Example A. 2 < 4, –3< –2, 0 > –1 are true statements
and –2 < –5 , 5 < 3 are false statements.
If we want all the numbers greater than 5, we may denote them
as "all number x where 5 < x". We write "a < x" for all the
numbers x greater than a, but not including a. In picture,
+–
a
open dot
If we want all the numbers x greater than or equal to a
(including a), we write it as a < x. In picture
+–
a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all
the numbers x between a and b. A line segment as such is
called an interval.
+–
a a < x < b b
The Number Line

26. Example B.
a. Draw –1 < x < 3.
The Number Line

27. Example B.
a. Draw –1 < x < 3.
It’s in the natural form.
The Number Line

28. Example B.
a. Draw –1 < x < 3.
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
The Number Line

29. Example B.
a. Draw –1 < x < 3.
0 3
+
-1
–
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
The Number Line
x

30. Example B.
a. Draw –1 < x < 3.
0 3
+
-1
–
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
The Number Line
–1 ≤ x < 3

31. Example B.
a. Draw –1 < x < 3.
0 3
+
-1
–
b. Draw 0 > x > –3
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
The Number Line
–1 ≤ x < 3

32. Example B.
a. Draw –1 < x < 3.
0 3
+
-1
–
b. Draw 0 > x > –3
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
The Number Line
–1 ≤ x < 3

33. Example B.
a. Draw –1 < x < 3.
0 3
+
-1
–
b. Draw 0 > x > –3
0
+
-3
–
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
The Number Line
–1 ≤ x < 3

34. Example B.
a. Draw –1 < x < 3.
0 3
+
-1
–
b. Draw 0 > x > –3
0
+
-3
–
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
The Number Line
–1 ≤ x < 3
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
–3 < x < 0

35. Example B.
a. Draw –1 < x < 3.
0 3
+
-1
–
b. Draw 0 > x > –3
0
+
-3
–
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any
solution meaning that there isn’t any number that would fit the
description hence there is nothing to draw.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
The Number Line
–1 ≤ x < 3
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
–3 < x < 0

36. Example B.
a. Draw –1 < x < 3.
0 3
+
-1
–
b. Draw 0 > x > –3
0
+
-3
–
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any
solution meaning that there isn’t any number that would fit the
description hence there is nothing to draw.
Put it in the natural form –3 < x < 0.
Then mark the numbers and x in order accordingly.
The Number Line
–1 ≤ x < 3
It’s in the natural form. Mark the numbers and x on the line
in order accordingly.
–3 < x < 0
The number line converts numbers to picture and in order for
the pictures to be helpful, certain accuracy is required when
they are drawn by hand.

37. Following are two skills for drawing and scaling a line segment.
The Number Line

38. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
The Number Line

39. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
The Number Line

40. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line

41. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line

42. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two.

43. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two.

44. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two.

45. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two. Each small segment is 1/4 of the original.

46. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two. Each small segment is 1/4 of the original.
To cut a line segment K into 6 pieces, cut K in half, then cut
each half into 3 pieces.
K

47. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two. Each small segment is 1/4 of the original.
To cut a line segment K into 6 pieces, cut K in half, then cut
each half into 3 pieces.
K

48. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two. Each small segment is 1/4 of the original.
To cut a line segment K into 6 pieces, cut K in half, then cut
each half into 3 pieces.
K

49. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two. Each small segment is 1/4 of the original.
To cut a line segment K into 6 pieces, cut K in half, then cut
each half into 3 pieces. Each smaller segment is 1/6 of K.
K

50. Following are two skills for drawing and scaling a line segment.
* Find the midpoint that cuts the segment in two equal pieces.
* Find the two points that cut the segment in three equal pieces.
The Number Line
To cut a line segment into 4 pieces, cut it in half, then cut each
half into two. Each small segment is 1/4 of the original.
To cut a line segment K into 6 pieces, cut K in half, then cut
each half into 3 pieces. Each smaller segment is 1/6 of K.
K
If we divide each segment into two again, we would have
12 segments which may represent a ruler of one foot divided
into 12 inches.

51. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers.

52. The Number Line
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers.

53. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.

54. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first:

55. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.

56. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable.

57. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
0o

58. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
Draw two markers close to the two ends and label them
as ±40.
0o
40o
–40o

59. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
Draw two markers close to the two ends and label them
as ±40. Divide each segment into fourths for ±10,
±20, and ±30.
0o
40o
–40o

60. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
Draw two markers close to the two ends and label them
as ±40. Divide each segment into fourths for ±10,
±20, and ±30.
0o
40o
–40o

61. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
Draw two markers close to the two ends and label them
as ±40. Divide each segment into fourths for ±10,
±20, and ±30.
0o
40o
–40o

62. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
Draw two markers close to the two ends and label them
as ±40. Divide each segment into fourths for ±10,
±20, and ±30.
0o
40o
–40o
20o
–20o
10o
30o
–10o
–30o

63. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
Draw two markers close to the two ends and label them
as ±40. Divide each segment into fourths for ±10,
±20, and ±30. Use this scale to plot the numbers to
obtain a reasonable picture as shown.
0o
40o
–40o
20o
–20o
10o
30o
–10o
–30o
–40o

64. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
Draw two markers close to the two ends and label them
as ±40. Divide each segment into fourths for ±10,
±20, and ±30. Use this scale to plot the numbers to
obtain a reasonable picture as shown.
0o
40o
–40o
20o
–20o
10o
30o
–10o
–30o
–40o
–25o

65. The Number Line
To plot a list of numbers on a number line, first select a suitable
scale based on the numbers. For example, based on the list,
we may set the size between two markers on the line
to be 5, or 10, or 50, or 100, etc.. for easier plotting,
Example C. We record the following temperatures
during the year: 35o, –40o, 27o, –25o, 16o, 21o.
Draw a vertical scale with appropriate spacing
representing temperature then plot these numbers.
Order the numbers first: –40, –25, 16, 21, 27, and 35.
The furthest we need to plot from the origin is –40
hence using 10 as the spacing between the markers is
reasonable. Draw a line and label its center as 0.
Draw two markers close to the two ends and label them
as ±40. Divide each segment into fourths for ±10,
±20, and ±30. Use this scale to plot the numbers to
obtain a reasonable picture as shown.
0o
40o
–40o
20o
–20o
10o
30o
–10o
–30o
35o
–40o
–25o
16o
21o
27o

66. The Number Line
Here are two important formulas about the number line.

67. The Number Line
Here are two important formulas about the number line.
Using a ruler we compute the length of a stick S by subtraction.
S

68. The Number Line
Ruler
Here are two important formulas about the number line.
Using a ruler we compute the length of a stick S by subtraction.
3
S
44

69. The Number Line
Ruler
Here are two important formulas about the number line.
Using a ruler we compute the length of a stick S by subtraction.
For example, the length of S
shown here is 44 – 3 = 41
which is the also distance from one end to the other.
3
S
44

70. The Number Line
Ruler
Here are two important formulas about the number line.
Using a ruler we compute the length of a stick S by subtraction.
For example, the length of S
shown here is 44 – 3 = 41
which is the also distance from one end to the other.
3
S
I. The Distance Formula.
The distance between two positions on the number line is
R – L where R is number to the right and L is number to the left.
44

71. The Number Line
Ruler
Here are two important formulas about the number line.
Example D.
a. Town A and town B are as shown on a map. What is the
distance between them?
Using a ruler we compute the length of a stick S by subtraction.
For example, the length of S
shown here is 44 – 3 = 41
which is the also distance from one end to the other.
3
S
I. The Distance Formula.
The distance between two positions on the number line is
R – L where R is number to the right and L is number to the left.
35 mi 97 mA B
44
0

72. The Number Line
Ruler
Here are two important formulas about the number line.
Example D.
a. Town A and town B are as shown on a map. What is the
distance between them?
Using a ruler we compute the length of a stick S by subtraction.
For example, the length of S
shown here is 44 – 3 = 41
which is the also distance from one end to the other.
3
S
I. The Distance Formula.
The distance between two positions on the number line is
R – L where R is number to the right and L is number to the left.
35 mi 97 mA B
44
0
The distance between them is 97 – 35 = 62 miles.

73. The Number Line
Ruler
Here are two important formulas about the number line.
Example D.
a. Town A and town B are as shown on a map. What is the
distance between them?
Using a ruler we compute the length of a stick S by subtraction.
For example, the length of S
shown here is 44 – 3 = 41
which is the also distance from one end to the other.
3
S
I. The Distance Formula.
The distance between two positions on the number line is
R – L where R is number to the right and L is number to the left.
35 mi 97 mA B
44
0
The distance between them is 97 – 35 = 62 miles.
b. What is the distance between the points u = –3 and v = 25?
u v
–3 25
0

74. The Number Line
Ruler
Here are two important formulas about the number line.
Example D.
a. Town A and town B are as shown on a map. What is the
distance between them?
Using a ruler we compute the length of a stick S by subtraction.
For example, the length of S
shown here is 44 – 3 = 41
which is the also distance from one end to the other.
3
S
I. The Distance Formula.
The distance between two positions on the number line is
R – L where R is number to the right and L is number to the left.
35 mi 97 mA B
44
0
The distance between them is 97 – 35 = 62 miles.
b. What is the distance between the points u = –3 and v = 25?
The point v = 25 is to the right of u = –3,
so the distance is the 25 – (–3) = 28.
u v
–3 25
0

75. The Number Line
Ruler
Here are two important formulas about the number line.
Example D.
a. Town A and town B are as shown on a map. What is the
distance between them?
Using a ruler we compute the length of a stick S by subtraction.
For example, the length of S
shown here is 44 – 3 = 41
which is the also distance from one end to the other.
3
S
I. The Distance Formula.
The distance between two positions on the number line is
R – L where R is number to the right and L is number to the left.
35 mi 97 mA B
44
0
The distance between them is 97 – 35 = 62 miles.
b. What is the distance between the points u = –3 and v = 25?
The point v = 25 is to the right of u = –3,
so the distance is the 25 – (–3) = 28. R – L = 28
u v
–3 25
0

76. The Number Line
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,

77. The Number Line
a
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
b

78. The Number Line
a a + b
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
b
2
the midpoint

79. The Number Line
a a + b b
2
the midpoint
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
this is also the average of a and b.

80. The Number Line
Example D.
a. Joe obtained 4 points on the 1st quiz and 7 points on the 2nd
quiz, what is the average of the two quizzes? Draw.
a a + b b
2
the midpoint
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
this is also the average of a and b.

81. The Number Line
Example D.
a. Joe obtained 4 points on the 1st quiz and 7 points on the 2nd
quiz, what is the average of the two quizzes? Draw.
a a + b b
The average of the two quizzes is
(4 + 7)/2 = 11/ 2 = 5.5
which is the midpoint of 4 and 7.
2
the midpoint
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
this is also the average of a and b.

82. The Number Line
Example D.
a. Joe obtained 4 points on the 1st quiz and 7 points on the 2nd
quiz, what is the average of the two quizzes? Draw.
a a + b
4
b
The average of the two quizzes is
(4 + 7)/2 = 11/ 2 = 5.5
which is the midpoint of 4 and 7.
2
the midpoint
7
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
this is also the average of a and b.

83. The Number Line
Example D.
a. Joe obtained 4 points on the 1st quiz and 7 points on the 2nd
quiz, what is the average of the two quizzes? Draw.
a a + b
4
b
the midpointThe average of the two quizzes is
(4 + 7)/2 = 11/ 2 = 5.5
which is the midpoint of 4 and 7.
2
the midpoint
7
5.5
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
this is also the average of a and b.

84. The Number Line
Example D.
a. Joe obtained 4 points on the 1st quiz and 7 points on the 2nd
quiz, what is the average of the two quizzes? Draw.
a a + b
4
b
the midpointThe average of the two quizzes is
(4 + 7)/2 = 11/ 2 = 5.5
which is the midpoint of 4 and 7.
b. Find the midpoints between u = –3 and v = 25?
2
the midpoint
7
5.5
–3 0 25
the midpoint
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
this is also the average of a and b.

85. The Number Line
Example D.
a. Joe obtained 4 points on the 1st quiz and 7 points on the 2nd
quiz, what is the average of the two quizzes? Draw.
a a + b
4
b
the midpointThe average of the two quizzes is
(4 + 7)/2 = 11/ 2 = 5.5
which is the midpoint of 4 and 7.
b. Find the midpoints between u = –3 and v = 25?
Their midpoint is (25 + (–3))/2 = 22/2 = 11.
2
the midpoint
7
5.5
–3 0 25
11
the midpoint
II. The Midpoint Formula.
The midpoint between two points a and b is (a + b)/2,
this is also the average of a and b.

86. Exercise. A. Draw the following Inequalities. Indicate clearly
whether the end points are included or not.
1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12
B. Write in the natural form then draw them.
5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12
C. Draw the following intervals, state so if it is impossible.
9. 6 > x ≥ 3 10. –5 < x ≤ 2 11. 1 > x ≥ –8 12. 4 < x ≤ 2
13. 6 > x ≥ 8 14. –5 > x ≤ 2 15. –7 ≤ x ≤ –3 16. –7 ≤ x ≤ –9
D. Solve the following Inequalities and draw the solution.
17. x + 5 < 3 18. –5 ≤ 2x + 3 19. 3x + 1 < –8
20. 2x + 3 ≤ 12 – x 21. –3x + 5 ≥ 1 – 4x
22. 2(x + 2) ≥ 5 – (x – 1) 23. 3(x – 1) + 2 ≤ – 2x – 9
24. –2(x – 3) > 2(–x – 1) + 3x 25. –(x + 4) – 2 ≤ 4(x – 1)
26. x + 2(x – 3) < 2(x – 1) – 2
27. –2(x – 3) + 3 ≥ 2(x – 1) + 3x + 13