← Inclusive vs exclusive range boundaries · Inequality Range Membership

Inclusive vs exclusive range boundaries · 12 practice problems

6.EE.B.86.NS.C.6

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Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy

On the number line, the shaded range contains exactly $7$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $18$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $18$ there is an open circle, meaning $18$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 18, which is not. Exactly 7 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 18".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 18 does not count.
The range holds 7 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 18 means 18 is outside, so the largest number in the range is 17.

A \le n \le 17

The range runs from A up to 17, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 11: the whole numbers would be 11, 12, 13, ..., 17. Counting them inclusively gives 17 minus 11 plus 1.

17 - 11 + 1 = 7

That is exactly the 7 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 18 is not, the count is simply 18 minus A -- no plus one, no minus one.

18 - A = 7 \Rightarrow A = 11

A is 11.

Answer: 11

4 · Reviewdoes it hold up?

Check the neighbours: starting at 10 would give 8 numbers and starting at 12 would give 6. Only 11 gives 7.

Another way: Counting the other way round: 17 is the last member and 7 members counted back from it lands on 11.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 2 easy

On the number line, the shaded range contains exactly $9$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $25$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $25$ there is an open circle, meaning $25$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 25, which is not. Exactly 9 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 25".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 25 does not count.
The range holds 9 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 25 means 25 is outside, so the largest number in the range is 24.

A \le n \le 24

The range runs from A up to 24, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 16: the whole numbers would be 16, 17, 18, ..., 24. Counting them inclusively gives 24 minus 16 plus 1.

24 - 16 + 1 = 9

That is exactly the 9 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 25 is not, the count is simply 25 minus A -- no plus one, no minus one.

25 - A = 9 \Rightarrow A = 16

A is 16.

Answer: 16

4 · Reviewdoes it hold up?

Check the neighbours: starting at 15 would give 10 numbers and starting at 17 would give 8. Only 16 gives 9.

Another way: Counting the other way round: 24 is the last member and 9 members counted back from it lands on 16.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 3 easy

On the number line, the shaded range contains exactly $11$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $27$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $27$ there is an open circle, meaning $27$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 27, which is not. Exactly 11 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 27".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 27 does not count.
The range holds 11 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 27 means 27 is outside, so the largest number in the range is 26.

A \le n \le 26

The range runs from A up to 26, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 16: the whole numbers would be 16, 17, 18, ..., 26. Counting them inclusively gives 26 minus 16 plus 1.

26 - 16 + 1 = 11

That is exactly the 11 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 27 is not, the count is simply 27 minus A -- no plus one, no minus one.

27 - A = 11 \Rightarrow A = 16

A is 16.

Answer: 16

4 · Reviewdoes it hold up?

Check the neighbours: starting at 15 would give 12 numbers and starting at 17 would give 10. Only 16 gives 11.

Another way: Counting the other way round: 26 is the last member and 11 members counted back from it lands on 16.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 4 easy

On the number line, the shaded range contains exactly $13$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $29$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $29$ there is an open circle, meaning $29$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 29, which is not. Exactly 13 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 29".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 29 does not count.
The range holds 13 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 29 means 29 is outside, so the largest number in the range is 28.

A \le n \le 28

The range runs from A up to 28, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 16: the whole numbers would be 16, 17, 18, ..., 28. Counting them inclusively gives 28 minus 16 plus 1.

28 - 16 + 1 = 13

That is exactly the 13 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 29 is not, the count is simply 29 minus A -- no plus one, no minus one.

29 - A = 13 \Rightarrow A = 16

A is 16.

Answer: 16

4 · Reviewdoes it hold up?

Check the neighbours: starting at 15 would give 14 numbers and starting at 17 would give 12. Only 16 gives 13.

Another way: Counting the other way round: 28 is the last member and 13 members counted back from it lands on 16.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 5 medium

On the number line, the shaded range contains exactly $15$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $33$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $33$ there is an open circle, meaning $33$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 33, which is not. Exactly 15 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 33".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 33 does not count.
The range holds 15 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 33 means 33 is outside, so the largest number in the range is 32.

A \le n \le 32

The range runs from A up to 32, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 18: the whole numbers would be 18, 19, 20, ..., 32. Counting them inclusively gives 32 minus 18 plus 1.

32 - 18 + 1 = 15

That is exactly the 15 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 33 is not, the count is simply 33 minus A -- no plus one, no minus one.

33 - A = 15 \Rightarrow A = 18

A is 18.

Answer: 18

4 · Reviewdoes it hold up?

Check the neighbours: starting at 17 would give 16 numbers and starting at 19 would give 14. Only 18 gives 15.

Another way: Counting the other way round: 32 is the last member and 15 members counted back from it lands on 18.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 6 medium

On the number line, the shaded range contains exactly $10$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $36$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $36$ there is an open circle, meaning $36$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 36, which is not. Exactly 10 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 36".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 36 does not count.
The range holds 10 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 36 means 36 is outside, so the largest number in the range is 35.

A \le n \le 35

The range runs from A up to 35, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 26: the whole numbers would be 26, 27, 28, ..., 35. Counting them inclusively gives 35 minus 26 plus 1.

35 - 26 + 1 = 10

That is exactly the 10 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 36 is not, the count is simply 36 minus A -- no plus one, no minus one.

36 - A = 10 \Rightarrow A = 26

A is 26.

Answer: 26

4 · Reviewdoes it hold up?

Check the neighbours: starting at 25 would give 11 numbers and starting at 27 would give 9. Only 26 gives 10.

Another way: Counting the other way round: 35 is the last member and 10 members counted back from it lands on 26.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 7 medium

On the number line, the shaded range contains exactly $12$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $40$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $40$ there is an open circle, meaning $40$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 40, which is not. Exactly 12 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 40".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 40 does not count.
The range holds 12 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 40 means 40 is outside, so the largest number in the range is 39.

A \le n \le 39

The range runs from A up to 39, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 28: the whole numbers would be 28, 29, 30, ..., 39. Counting them inclusively gives 39 minus 28 plus 1.

39 - 28 + 1 = 12

That is exactly the 12 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 40 is not, the count is simply 40 minus A -- no plus one, no minus one.

40 - A = 12 \Rightarrow A = 28

A is 28.

Answer: 28

4 · Reviewdoes it hold up?

Check the neighbours: starting at 27 would give 13 numbers and starting at 29 would give 11. Only 28 gives 12.

Another way: Counting the other way round: 39 is the last member and 12 members counted back from it lands on 28.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 8 medium

On the number line, the shaded range contains exactly $8$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $44$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $44$ there is an open circle, meaning $44$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 44, which is not. Exactly 8 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 44".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 44 does not count.
The range holds 8 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 44 means 44 is outside, so the largest number in the range is 43.

A \le n \le 43

The range runs from A up to 43, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 36: the whole numbers would be 36, 37, 38, ..., 43. Counting them inclusively gives 43 minus 36 plus 1.

43 - 36 + 1 = 8

That is exactly the 8 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 44 is not, the count is simply 44 minus A -- no plus one, no minus one.

44 - A = 8 \Rightarrow A = 36

A is 36.

Answer: 36

4 · Reviewdoes it hold up?

Check the neighbours: starting at 35 would give 9 numbers and starting at 37 would give 7. Only 36 gives 8.

Another way: Counting the other way round: 43 is the last member and 8 members counted back from it lands on 36.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 9 hard

On the number line, the shaded range contains exactly $20$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $52$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $52$ there is an open circle, meaning $52$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 52, which is not. Exactly 20 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 52".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 52 does not count.
The range holds 20 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 52 means 52 is outside, so the largest number in the range is 51.

A \le n \le 51

The range runs from A up to 51, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 32: the whole numbers would be 32, 33, 34, ..., 51. Counting them inclusively gives 51 minus 32 plus 1.

51 - 32 + 1 = 20

That is exactly the 20 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 52 is not, the count is simply 52 minus A -- no plus one, no minus one.

52 - A = 20 \Rightarrow A = 32

A is 32.

Answer: 32

4 · Reviewdoes it hold up?

Check the neighbours: starting at 31 would give 21 numbers and starting at 33 would give 19. Only 32 gives 20.

Another way: Counting the other way round: 51 is the last member and 20 members counted back from it lands on 32.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 10 hard

On the number line, the shaded range contains exactly $17$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $58$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $58$ there is an open circle, meaning $58$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 58, which is not. Exactly 17 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 58".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 58 does not count.
The range holds 17 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 58 means 58 is outside, so the largest number in the range is 57.

A \le n \le 57

The range runs from A up to 57, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 41: the whole numbers would be 41, 42, 43, ..., 57. Counting them inclusively gives 57 minus 41 plus 1.

57 - 41 + 1 = 17

That is exactly the 17 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 58 is not, the count is simply 58 minus A -- no plus one, no minus one.

58 - A = 17 \Rightarrow A = 41

A is 41.

Answer: 41

4 · Reviewdoes it hold up?

Check the neighbours: starting at 40 would give 18 numbers and starting at 42 would give 16. Only 41 gives 17.

Another way: Counting the other way round: 57 is the last member and 17 members counted back from it lands on 41.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 11 hard

On the number line, the shaded range contains exactly $14$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $61$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $61$ there is an open circle, meaning $61$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 61, which is not. Exactly 14 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 61".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 61 does not count.
The range holds 14 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 61 means 61 is outside, so the largest number in the range is 60.

A \le n \le 60

The range runs from A up to 60, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 47: the whole numbers would be 47, 48, 49, ..., 60. Counting them inclusively gives 60 minus 47 plus 1.

60 - 47 + 1 = 14

That is exactly the 14 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 61 is not, the count is simply 61 minus A -- no plus one, no minus one.

61 - A = 14 \Rightarrow A = 47

A is 47.

Answer: 47

4 · Reviewdoes it hold up?

Check the neighbours: starting at 46 would give 15 numbers and starting at 48 would give 13. Only 47 gives 14.

Another way: Counting the other way round: 60 is the last member and 14 members counted back from it lands on 47.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Variant 12 hard

On the number line, the shaded range contains exactly $25$ whole numbers. Find the whole number that belongs at $Ⓐ$ .

The range shown on the number line is " $Ⓐ$ or greater, and less than $75$ ."

Number line: at the left position $Ⓐ$ there is a filled (solid) dot, meaning $Ⓐ$ is included; at the right position $75$ there is an open circle, meaning $75$ is not included. The segment between the two points is drawn in bold.

Your answer

Show solution

1 · Understandwhat's really being asked

A stretch of the number line is shaded. It starts at an unknown whole number, which is included, and stops before 75, which is not. Exactly 25 whole numbers lie in it. We need the starting number.

Givens

The range is "the unknown or greater, and less than 75".
The left dot is filled, so the left endpoint counts.
The right circle is open, so 75 does not count.
The range holds 25 whole numbers.

Unknowns

The whole number at the left end.

Constraints

Only whole numbers are counted.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #1 Draw a Diagram#2 Make a Systematic List

Read the two dots first -- one included, one not -- because that is what decides how the count relates to the endpoints. Then a guess and a count confirm it.

3 · Execute3 carry out the plan

1Read the boundary dots

#1 Draw a Diagram 6.EE.B.8

The filled dot on the left means that number is inside the range. The open circle at 75 means 75 is outside, so the largest number in the range is 74.

A \le n \le 74

The range runs from A up to 74, both counted.

2Guess a value and count

#2 Make a Systematic List 6.NS.C.6

Try 50: the whole numbers would be 50, 51, 52, ..., 74. Counting them inclusively gives 74 minus 50 plus 1.

74 - 50 + 1 = 25

That is exactly the 25 the problem asks for.

3Confirm with the counting rule

#6 Guess and Check 6.EE.B.8

Because the left end is included and 75 is not, the count is simply 75 minus A -- no plus one, no minus one.

75 - A = 25 \Rightarrow A = 50

A is 50.

Answer: 50

4 · Reviewdoes it hold up?

Check the neighbours: starting at 49 would give 26 numbers and starting at 51 would give 24. Only 50 gives 25.

Another way: Counting the other way round: 74 is the last member and 25 members counted back from it lands on 50.

Standardsmin grade 6

6.EE.B.8 Write an inequality of the form x > c or x < c and graph on a number line — Reading the filled and open dots as an inequality with one endpoint included.
6.NS.C.6 Understand a rational number as a point on the number line — Counting the whole numbers between the two marked points.

💡Takeaway. A filled dot counts and an open one does not -- that single difference is what decides whether you add one or not.

Where next?

← Back to the problem: Inclusive vs exclusive range boundaries