53 the real line

The Number Line

Back to Algebra–Ready Review Content.

The Number LineJust like assigning address to houses on a street we assign

addresses to points on a line.

The Number LineJust 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

The Number LineJust 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

The Number LineJust 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

The Number LineJust 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

The Number LineJust 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

The Number LineJust 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

The Number LineJust 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

The Number LineJust 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

The Number LineJust 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<

The Number LineJust 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<

The Number LineJust 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<

Example A. 2 < 4, –3< –2, 0 > –1 are true statements

The Number Line

Example A. 2 < 4, –3< –2, 0 > –1 are true statements

and –2 < –5 , 5 < 3 are false statements.

The Number Line

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

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

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

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

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

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

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

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

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

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

Example B.

a. Draw –1 < x < 3.

The Number Line

Example B.

a. Draw –1 < x < 3.

It’s in the natural form.

The Number Line

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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.

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.

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.

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.

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

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

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

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

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.

The Number LineTo plot a list of numbers on a number line, first select a suitable

scale based on the numbers.

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.

The Number LineTo 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.

The Number LineTo 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:

The Number LineTo 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 Number LineTo 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.

The Number LineTo 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

The Number LineTo 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

The Number LineTo 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

The Number LineTo 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

The Number LineTo 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

The Number LineTo 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

The Number LineTo 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

The Number LineTo 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

The Number LineTo 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

The Number LineFollowing are two basic formulas about the number line.

The Number Line

Ruler

Following are two basic formulas about the number line.

Using a ruler we compute the length of a stick S by subtraction.

3

S

44

The Number Line

Ruler

Following are two basic 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 end to end.3

S

44

The Number Line

Ruler

Following are two basic 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 end to end.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

The Number Line

Ruler

Following are two basic 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 end to end.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 Number Line

Ruler

Following are two basic 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 end to end.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.

The Number Line

Ruler

Following are two basic 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 end to end.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

The Number Line

Ruler

Following are two basic 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 end to end.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

The Number Line

a a + b

II. The Midpoint Formula.

The midpoint between two points a and b on the number line is

(a + b)/2, this is also the average of a and b.

b2

the midpoint

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

II. The Midpoint Formula.

The midpoint between two points a and b on the number line is

(a + b)/2, this is also the average of a and b.

b2

the midpoint

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

II. The Midpoint Formula.

The midpoint between two points a and b on the number line is

(a + b)/2, this is also the average of a and 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

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

II. The Midpoint Formula.

The midpoint between two points a and b on the number line is

(a + b)/2, this is also the average of a and 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

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

II. The Midpoint Formula.

The midpoint between two points a and b on the number line is

(a + b)/2, this is also the average of a and b.

4

b

the midpoint 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

75.5

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

II. The Midpoint Formula.

The midpoint between two points a and b on the number line is

(a + b)/2, this is also the average of a and b.

4

b

the midpoint The 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

75.5

–3 0 25the midpoint

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

II. The Midpoint Formula.

The midpoint between two points a and b on the number line is

(a + b)/2, this is also the average of a and b.

4

b

the midpoint The 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

75.5

–3 0 25

11

the midpoint

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