The Number Line

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”.

0L 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.

0L R<

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

Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.

The Number Line

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

KIf 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 lineto 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 LineHere are two important formulas about the number line.

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

S

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

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 = 41which is the also distance from one end to the other.

3

S

44

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 = 41which 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

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 = 41which 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 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 = 41which 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.

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 = 41which 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

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 = 41which 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

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 = 41which 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

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

The Number Line

a

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

b

The Number Line

a a + b

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

b2

the midpoint

The Number Line

a a + b b2

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.

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 b2

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.

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.

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.

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 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

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.

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 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

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.

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 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 2511

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.

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 ≤ 213. 6 > x ≥ 8

14. –5 > x ≤ 2 15. –7 ≤ x ≤ –3 16. –7 ≤ x ≤ –9D. 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 – 924. –2(x – 3) > 2(–x – 1) + 3x 25. –(x + 4) – 2 ≤ 4(x – 1)26. x + 2(x – 3) < 2(x – 1) – 227. –2(x – 3) + 3 ≥ 2(x – 1) + 3x + 13