Practice · Pin the boundary so exactly three naturals fit · Sensim Math

Variant 1 easy

The inequality $\frac{5x-1}{2}-a\le x+3$ has exactly $3$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly three natural numbers lie at or below that boundary.

Givens

$\frac{5x-1}{2}-a\le x+3$
Exactly $3$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

2

and gather terms.

5x-1-2a\le 2x+6 \;\Rightarrow\; 3x\le 2a+7 \;\Rightarrow\; x\le\frac{2a+7}{3}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3

, the boundary must reach

3

(so

3

is included) but stay below

4

(so

4

is excluded).

3\le\frac{2a+7}{3}<4

The boundary may equal

3

because the sign is

\le

; it may not equal

4

, or

4

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

3

, subtract

7

, divide by

2

.

9\le 2a+7<12 \;\Rightarrow\; 2\le 2a<5 \;\Rightarrow\; 1\le a<\frac{5}{2}

Applying the same operation to all three parts keeps the chain true.

Answer:

1\le a<\frac{5}{2}

4 · Reviewdoes it hold up?

At $a=1$ the solution set is $x\le 3$ : naturals $1, 2, 3$ , exactly $3$ . At $a=\frac{5}{2}$ it is $x\le 4$ , giving $4$ naturals, so $\frac{5}{2}$ must be excluded. For $a$ a little below $1$ the boundary drops below $3$ , leaving only $2$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $2$ to $3$ exactly when the boundary reaches $3$ ( $a=1$ ) and from $3$ to $4$ when it reaches $4$ ( $a=\frac{5}{2}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 2 easy

The inequality $\frac{5x+3}{2}-a\le x-1$ has exactly $4$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly $4$ natural numbers lie at or below that boundary.

Givens

$\frac{5x+3}{2}-a\le x-1$
Exactly $4$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3, 4$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

2

and gather terms.

5x+3-2a\le 2x-2 \;\Rightarrow\; 3x\le 2a-5 \;\Rightarrow\; x\le\frac{2a-5}{3}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3, 4

, the boundary must reach

4

(so

4

is included) but stay below

5

(so

5

is excluded).

4\le\frac{2a-5}{3}<5

The boundary may equal

4

because the sign is

\le

; it may not equal

5

, or

5

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

3

, add

5

, divide by

2

.

12\le 2a-5<15 \;\Rightarrow\; 17\le 2a<20 \;\Rightarrow\; \frac{17}{2}\le a<10

Applying the same operation to all three parts keeps the chain true.

Answer:

\frac{17}{2}\le a<10

4 · Reviewdoes it hold up?

At $a=\frac{17}{2}$ the solution set is $x\le 4$ : naturals $1, 2, 3, 4$ , exactly $4$ . At $a=10$ it is $x\le 5$ , giving $5$ naturals, so $10$ must be excluded. For $a$ a little below $\frac{17}{2}$ the boundary drops below $4$ , leaving only $3$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $3$ to $4$ exactly when the boundary reaches $4$ ( $a=\frac{17}{2}$ ) and from $4$ to $5$ when it reaches $5$ ( $a=10$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 3 easy

The inequality $\frac{5x-2}{3}-a\le x+2$ has exactly $4$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly $4$ natural numbers lie at or below that boundary.

Givens

$\frac{5x-2}{3}-a\le x+2$
Exactly $4$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3, 4$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

3

and gather terms.

5x-2-3a\le 3x+6 \;\Rightarrow\; 2x\le 3a+8 \;\Rightarrow\; x\le\frac{3a+8}{2}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3, 4

, the boundary must reach

4

(so

4

is included) but stay below

5

(so

5

is excluded).

4\le\frac{3a+8}{2}<5

The boundary may equal

4

because the sign is

\le

; it may not equal

5

, or

5

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

2

, subtract

8

, divide by

3

.

8\le 3a+8<10 \;\Rightarrow\; 0\le 3a<2 \;\Rightarrow\; 0\le a<\frac{2}{3}

Applying the same operation to all three parts keeps the chain true.

Answer:

0\le a<\frac{2}{3}

4 · Reviewdoes it hold up?

At $a=0$ the solution set is $x\le 4$ : naturals $1, 2, 3, 4$ , exactly $4$ . At $a=\frac{2}{3}$ it is $x\le 5$ , giving $5$ naturals, so $\frac{2}{3}$ must be excluded. For $a$ a little below $0$ the boundary drops below $4$ , leaving only $3$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $3$ to $4$ exactly when the boundary reaches $4$ ( $a=0$ ) and from $4$ to $5$ when it reaches $5$ ( $a=\frac{2}{3}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 4 medium

The inequality $\frac{7x+4}{3}-a\le x-1$ has exactly $3$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly three natural numbers lie at or below that boundary.

Givens

$\frac{7x+4}{3}-a\le x-1$
Exactly $3$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

3

and gather terms.

7x+4-3a\le 3x-3 \;\Rightarrow\; 4x\le 3a-7 \;\Rightarrow\; x\le\frac{3a-7}{4}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3

, the boundary must reach

3

(so

3

is included) but stay below

4

(so

4

is excluded).

3\le\frac{3a-7}{4}<4

The boundary may equal

3

because the sign is

\le

; it may not equal

4

, or

4

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

4

, add

7

, divide by

3

.

12\le 3a-7<16 \;\Rightarrow\; 19\le 3a<23 \;\Rightarrow\; \frac{19}{3}\le a<\frac{23}{3}

Applying the same operation to all three parts keeps the chain true.

Answer:

\frac{19}{3}\le a<\frac{23}{3}

4 · Reviewdoes it hold up?

At $a=\frac{19}{3}$ the solution set is $x\le 3$ : naturals $1, 2, 3$ , exactly $3$ . At $a=\frac{23}{3}$ it is $x\le 4$ , giving $4$ naturals, so $\frac{23}{3}$ must be excluded. For $a$ a little below $\frac{19}{3}$ the boundary drops below $3$ , leaving only $2$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $2$ to $3$ exactly when the boundary reaches $3$ ( $a=\frac{19}{3}$ ) and from $3$ to $4$ when it reaches $4$ ( $a=\frac{23}{3}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 5 easy

The inequality $\frac{7x-2}{3}-a\le x+1$ has exactly $3$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly three natural numbers lie at or below that boundary.

Givens

$\frac{7x-2}{3}-a\le x+1$
Exactly $3$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

3

and gather terms.

7x-2-3a\le 3x+3 \;\Rightarrow\; 4x\le 3a+5 \;\Rightarrow\; x\le\frac{3a+5}{4}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3

, the boundary must reach

3

(so

3

is included) but stay below

4

(so

4

is excluded).

3\le\frac{3a+5}{4}<4

The boundary may equal

3

because the sign is

\le

; it may not equal

4

, or

4

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

4

, subtract

5

, divide by

3

.

12\le 3a+5<16 \;\Rightarrow\; 7\le 3a<11 \;\Rightarrow\; \frac{7}{3}\le a<\frac{11}{3}

Applying the same operation to all three parts keeps the chain true.

Answer:

\frac{7}{3}\le a<\frac{11}{3}

4 · Reviewdoes it hold up?

At $a=\frac{7}{3}$ the solution set is $x\le 3$ : naturals $1, 2, 3$ , exactly $3$ . At $a=\frac{11}{3}$ it is $x\le 4$ , giving $4$ naturals, so $\frac{11}{3}$ must be excluded. For $a$ a little below $\frac{7}{3}$ the boundary drops below $3$ , leaving only $2$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $2$ to $3$ exactly when the boundary reaches $3$ ( $a=\frac{7}{3}$ ) and from $3$ to $4$ when it reaches $4$ ( $a=\frac{11}{3}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 6 medium

The inequality $\frac{7x-3}{2}-a\le 2x+4$ has exactly $5$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly $5$ natural numbers lie at or below that boundary.

Givens

$\frac{7x-3}{2}-a\le 2x+4$
Exactly $5$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3, 4, 5$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

2

and gather terms.

7x-3-2a\le 4x+8 \;\Rightarrow\; 3x\le 2a+11 \;\Rightarrow\; x\le\frac{2a+11}{3}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3, 4, 5

, the boundary must reach

5

(so

5

is included) but stay below

6

(so

6

is excluded).

5\le\frac{2a+11}{3}<6

The boundary may equal

5

because the sign is

\le

; it may not equal

6

, or

6

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

3

, subtract

11

, divide by

2

.

15\le 2a+11<18 \;\Rightarrow\; 4\le 2a<7 \;\Rightarrow\; 2\le a<\frac{7}{2}

Applying the same operation to all three parts keeps the chain true.

Answer:

2\le a<\frac{7}{2}

4 · Reviewdoes it hold up?

At $a=2$ the solution set is $x\le 5$ : naturals $1, 2, 3, 4, 5$ , exactly $5$ . At $a=\frac{7}{2}$ it is $x\le 6$ , giving $6$ naturals, so $\frac{7}{2}$ must be excluded. For $a$ a little below $2$ the boundary drops below $5$ , leaving only $4$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $4$ to $5$ exactly when the boundary reaches $5$ ( $a=2$ ) and from $5$ to $6$ when it reaches $6$ ( $a=\frac{7}{2}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 7 medium

The inequality $\frac{7x+1}{2}-a\le 2x+1$ has exactly $2$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly $2$ natural numbers lie at or below that boundary.

Givens

$\frac{7x+1}{2}-a\le 2x+1$
Exactly $2$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

2

and gather terms.

7x+1-2a\le 4x+2 \;\Rightarrow\; 3x\le 2a+1 \;\Rightarrow\; x\le\frac{2a+1}{3}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2

, the boundary must reach

2

(so

2

is included) but stay below

3

(so

3

is excluded).

2\le\frac{2a+1}{3}<3

The boundary may equal

2

because the sign is

\le

; it may not equal

3

, or

3

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

3

, subtract

1

, divide by

2

.

6\le 2a+1<9 \;\Rightarrow\; 5\le 2a<8 \;\Rightarrow\; \frac{5}{2}\le a<4

Applying the same operation to all three parts keeps the chain true.

Answer:

\frac{5}{2}\le a<4

4 · Reviewdoes it hold up?

At $a=\frac{5}{2}$ the solution set is $x\le 2$ : naturals $1, 2$ , exactly $2$ . At $a=4$ it is $x\le 3$ , giving $3$ naturals, so $4$ must be excluded. For $a$ a little below $\frac{5}{2}$ the boundary drops below $2$ , leaving only $1$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $1$ to $2$ exactly when the boundary reaches $2$ ( $a=\frac{5}{2}$ ) and from $2$ to $3$ when it reaches $3$ ( $a=4$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 8 medium

The inequality $\frac{7x+1}{2}-a\le 2x-3$ has exactly $4$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly $4$ natural numbers lie at or below that boundary.

Givens

$\frac{7x+1}{2}-a\le 2x-3$
Exactly $4$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3, 4$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

2

and gather terms.

7x+1-2a\le 4x-6 \;\Rightarrow\; 3x\le 2a-7 \;\Rightarrow\; x\le\frac{2a-7}{3}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3, 4

, the boundary must reach

4

(so

4

is included) but stay below

5

(so

5

is excluded).

4\le\frac{2a-7}{3}<5

The boundary may equal

4

because the sign is

\le

; it may not equal

5

, or

5

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

3

, add

7

, divide by

2

.

12\le 2a-7<15 \;\Rightarrow\; 19\le 2a<22 \;\Rightarrow\; \frac{19}{2}\le a<11

Applying the same operation to all three parts keeps the chain true.

Answer:

\frac{19}{2}\le a<11

4 · Reviewdoes it hold up?

At $a=\frac{19}{2}$ the solution set is $x\le 4$ : naturals $1, 2, 3, 4$ , exactly $4$ . At $a=11$ it is $x\le 5$ , giving $5$ naturals, so $11$ must be excluded. For $a$ a little below $\frac{19}{2}$ the boundary drops below $4$ , leaving only $3$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $3$ to $4$ exactly when the boundary reaches $4$ ( $a=\frac{19}{2}$ ) and from $4$ to $5$ when it reaches $5$ ( $a=11$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 9 hard

The inequality $\frac{8x-1}{3}-a\le 2x+2$ has exactly $2$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly $2$ natural numbers lie at or below that boundary.

Givens

$\frac{8x-1}{3}-a\le 2x+2$
Exactly $2$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

3

and gather terms.

8x-1-3a\le 6x+6 \;\Rightarrow\; 2x\le 3a+7 \;\Rightarrow\; x\le\frac{3a+7}{2}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2

, the boundary must reach

2

(so

2

is included) but stay below

3

(so

3

is excluded).

2\le\frac{3a+7}{2}<3

The boundary may equal

2

because the sign is

\le

; it may not equal

3

, or

3

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

2

, subtract

7

, divide by

3

.

4\le 3a+7<6 \;\Rightarrow\; -3\le 3a<-1 \;\Rightarrow\; -1\le a<-\frac{1}{3}

Applying the same operation to all three parts keeps the chain true.

Answer:

-1\le a<-\frac{1}{3}

4 · Reviewdoes it hold up?

At $a=-1$ the solution set is $x\le 2$ : naturals $1, 2$ , exactly $2$ . At $a=-\frac{1}{3}$ it is $x\le 3$ , giving $3$ naturals, so $-\frac{1}{3}$ must be excluded. For $a$ a little below $-1$ the boundary drops below $2$ , leaving only $1$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $1$ to $2$ exactly when the boundary reaches $2$ ( $a=-1$ ) and from $2$ to $3$ when it reaches $3$ ( $a=-\frac{1}{3}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 10 hard

The inequality $\frac{9x+1}{2}-a\le 3x+1$ has exactly $3$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly three natural numbers lie at or below that boundary.

Givens

$\frac{9x+1}{2}-a\le 3x+1$
Exactly $3$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

2

and gather terms.

9x+1-2a\le 6x+2 \;\Rightarrow\; 3x\le 2a+1 \;\Rightarrow\; x\le\frac{2a+1}{3}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3

, the boundary must reach

3

(so

3

is included) but stay below

4

(so

4

is excluded).

3\le\frac{2a+1}{3}<4

The boundary may equal

3

because the sign is

\le

; it may not equal

4

, or

4

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

3

, subtract

1

, divide by

2

.

9\le 2a+1<12 \;\Rightarrow\; 8\le 2a<11 \;\Rightarrow\; 4\le a<\frac{11}{2}

Applying the same operation to all three parts keeps the chain true.

Answer:

4\le a<\frac{11}{2}

4 · Reviewdoes it hold up?

At $a=4$ the solution set is $x\le 3$ : naturals $1, 2, 3$ , exactly $3$ . At $a=\frac{11}{2}$ it is $x\le 4$ , giving $4$ naturals, so $\frac{11}{2}$ must be excluded. For $a$ a little below $4$ the boundary drops below $3$ , leaving only $2$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $2$ to $3$ exactly when the boundary reaches $3$ ( $a=4$ ) and from $3$ to $4$ when it reaches $4$ ( $a=\frac{11}{2}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 11 hard

The inequality $\frac{11x-1}{4}-a\le 2x+1$ has exactly $3$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly three natural numbers lie at or below that boundary.

Givens

$\frac{11x-1}{4}-a\le 2x+1$
Exactly $3$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2, 3$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

4

and gather terms.

11x-1-4a\le 8x+4 \;\Rightarrow\; 3x\le 4a+5 \;\Rightarrow\; x\le\frac{4a+5}{3}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2, 3

, the boundary must reach

3

(so

3

is included) but stay below

4

(so

4

is excluded).

3\le\frac{4a+5}{3}<4

The boundary may equal

3

because the sign is

\le

; it may not equal

4

, or

4

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

3

, subtract

5

, divide by

4

.

9\le 4a+5<12 \;\Rightarrow\; 4\le 4a<7 \;\Rightarrow\; 1\le a<\frac{7}{4}

Applying the same operation to all three parts keeps the chain true.

Answer:

1\le a<\frac{7}{4}

4 · Reviewdoes it hold up?

At $a=1$ the solution set is $x\le 3$ : naturals $1, 2, 3$ , exactly $3$ . At $a=\frac{7}{4}$ it is $x\le 4$ , giving $4$ naturals, so $\frac{7}{4}$ must be excluded. For $a$ a little below $1$ the boundary drops below $3$ , leaving only $2$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $2$ to $3$ exactly when the boundary reaches $3$ ( $a=1$ ) and from $3$ to $4$ when it reaches $4$ ( $a=\frac{7}{4}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Variant 12 hard

The inequality $\frac{11x+1}{3}-a\le 3x-2$ has exactly $2$ solutions that are natural numbers (positive integers). Find the range of values of the constant $a$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Solve the inequality for $x$ ; the answer is " $x$ is at most some boundary that depends on $a$ ". Choose $a$ so that exactly $2$ natural numbers lie at or below that boundary.

Givens

$\frac{11x+1}{3}-a\le 3x-2$
Exactly $2$ natural numbers satisfy it.

Unknowns

The range of $a$ .

Constraints

Natural numbers start at $1$ .
Watch which endpoint is included.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #1 Draw a Diagram

Turn the inequality into $x\le$ (boundary). Then place the boundary on a number line: the naturals below it must be exactly $1, 2$ .

3 · Execute3 carry out the plan

1Clear the fraction

#13 Convert to Algebra 7.EE.B.4

Multiply both sides by

3

and gather terms.

11x+1-3a\le 9x-6 \;\Rightarrow\; 2x\le 3a-7 \;\Rightarrow\; x\le\frac{3a-7}{2}

Multiplying by a positive number keeps the inequality direction.

2Place the boundary on a number line

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

The natural solutions are

1, 2, 3, \dots

up to the boundary. To get exactly

1, 2

, the boundary must reach

2

(so

2

is included) but stay below

3

(so

3

is excluded).

2\le\frac{3a-7}{2}<3

The boundary may equal

2

because the sign is

\le

; it may not equal

3

, or

3

would also be a solution.

3Solve for $a$

#13 Convert to Algebra 7.EE.B.4

Multiply every part by

2

, add

7

, divide by

3

.

4\le 3a-7<6 \;\Rightarrow\; 11\le 3a<13 \;\Rightarrow\; \frac{11}{3}\le a<\frac{13}{3}

Applying the same operation to all three parts keeps the chain true.

Answer:

\frac{11}{3}\le a<\frac{13}{3}

4 · Reviewdoes it hold up?

At $a=\frac{11}{3}$ the solution set is $x\le 2$ : naturals $1, 2$ , exactly $2$ . At $a=\frac{13}{3}$ it is $x\le 3$ , giving $3$ naturals, so $\frac{13}{3}$ must be excluded. For $a$ a little below $\frac{11}{3}$ the boundary drops below $2$ , leaving only $1$ naturals.

Another way: Guess and check endpoint values of $a$ : the count of natural solutions jumps from $1$ to $2$ exactly when the boundary reaches $2$ ( $a=\frac{11}{3}$ ) and from $2$ to $3$ when it reaches $3$ ( $a=\frac{13}{3}$ ).

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Solving the linear inequality and the chain inequality for $a$.
6.EE.B.8 Write and graph inequalities of the form x>c or x<c — Reading the solution $x\le$ (boundary) on a number line to count natural solutions.

💡Takeaway. Counting solutions is really about where the boundary sits between two whole numbers.

Pin the boundary so exactly three naturals fit · 12 practice problems

Generated variants — 12

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