PAPER · REASONING · PRACTICE

Inequality systems: overlap and feasible choices

Translate every restriction, take the overlap and then apply meaningful integer conditions.

40 minutes · Year 7, lower volume · text pp. 79–88 · textbook lessons 40–44

Textbook reference: Year 7 lower · Chapter 9 Inequalities and systems of inequalities · text lessons 40–44

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

18 minutes using tariffs: A=2n, B=10+n, minimum 8 visits, budget 25. Predict which plan is suitable at n=9 and n=14 (3 min). Calculate and record both price and feasibility, then solve each plan’s budget and minimum restrictions on paper (7 min). Hold all prices/minimum fixed and change only budget to 15; check whether either overlap remains (5 min). Explain why “cheaper” and “feasible” are different, with exact integer endpoints (3 min). The existing controls support these parameter values.

Organise the key idea

Translate every restriction, take the overlap and then apply meaningful integer conditions.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

A club pays a fixed 5 dollars plus 2 dollars per visit, has at most 27 dollars and needs at least 3 visits. Let x be the integer visit count. Form the restrictions and find all feasible counts.

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

Cost gives $2x+5\leq27$ , and need gives $x\geq3$ .
The cost limit is $x\leq11$ ; intersect to get $3\leq x\leq11$ .
Every integer from 3 through 11 is feasible; both extreme counts meet the budget.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Solve the system of inequalities; require every condition to hold.

\begin{cases}x>-1\\x\leq4\end{cases}

The first condition keeps values to the ____ of −1; that endpoint is open.
The second keeps values at or to the ____ of 4; that endpoint is filled.
Complete the overlap $-1\ \square\ x\ \square\ 4$ and draw it.

Use the stated conditions and the labelled scales.

x>−1 excludes −1 and extends right.
x≤4 includes 4 and extends left.
Only the overlapping region satisfies both; retain an open lower and closed upper endpoint.

Solve each original inequality separately, reversing direction only when dividing by a negative number.
Take their common part: $-1<x\leq 4$ .
Check every included boundary in the original inequalities.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite3

“x is at most 7” means which inequality?

$x\geq7$
$x<7$
$x\leq7$

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

At most allows 7 but excludes larger values.
Thus x≤7.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Is 2 included in $2<x\leq5$ ?

Yes; both endpoints are always included.
No; the lower inequality is strict.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

Substituting 2 gives 2<2, which is false.
Use an open lower endpoint.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Solve the system of inequalities; require every condition to hold.

\begin{cases}3x-2>4\\2x+1\leq11\end{cases}

Use the stated conditions and the labelled scales.

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

The first gives $x>2$ ; the second gives $x\leq5$ .
Both hold exactly on $2<x\leq5$ .

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice6

Solve the system of inequalities; require every condition to hold.

\begin{cases}x>4\\x\leq2\end{cases}

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

One condition requires x greater than 4, the other at most 2.
No real number meets both; the solution set is empty.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Notebooks cost 3 dollars each with one fixed 2-dollar wrapping fee. With 20 dollars, buy at least one notebook. Let x be the count; find all feasible counts.

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

The restrictions are $3x+2\leq20$ and $x\geq1$ .
Cost gives x≤6; integer counts are 1,2,3,4,5,6.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Solve the system of inequalities; require every condition to hold.

\begin{cases}x\geq3\\x\leq3\end{cases}

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

The number is at least 3 and at most 3.
Only x=3 meets both; the endpoint is included in each condition.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

List the integer solutions of $\frac{5}{2}<x<6$ .

\begin{cases}x>5/2\\x<6\end{cases}

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

Solve each original inequality separately, reversing direction only when dividing by a negative number.
Take their common part: $\frac{5}{2}<x<6$ .
Apply the stated integer restriction and check the extreme allowed integers.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

A costs 2n dollars; B costs 10+n dollars. For n=14 and budget 25 dollars, which is feasible?

Only B: 24 dollars; A costs 28.
Both, because B is cheaper.
Only A.

Separate the given information from what you must find.
Write the relevant definition or relation before calculating.
Check the result against every original condition, including units and endpoints.

A costs 28, exceeding 25.
B costs 24, within the budget.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Solve the system of inequalities; require every condition to hold.

\begin{cases}x\geq1\\x>-2\end{cases}

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

Solve each original inequality separately, reversing direction only when dividing by a negative number.
Take their common part: $x\geq1$ .
Check every included boundary in the original inequalities.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

An item costs 80 dollars. Let x be its selling price. The profit must be at least 15% of cost. Form an inequality and find the minimum price.

Collect unknown terms and constants using the same operation on both sides.
Decide whether the coefficient you divide by is positive or negative.
Reverse only for negative multiplication or division; then check strict versus included endpoints.

Profit divided by cost gives $\frac{x-80}{80}\geq\frac{15}{100}$ .
Multiply by positive 80: $x-80\geq12$ , so $x\geq92$ .
The minimum selling price is 92 dollars.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

A container has capacity 11 L, initially 3 L. Each full bottle adds 2 L. Let x be a nonnegative integer bottle count. Find all counts that fit.

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

Use $2x+3\leq11$ and $x\geq0$ .
Thus 0≤x≤4; counts 0 through 4 fit.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Solve the system of inequalities; require every condition to hold.

\begin{cases}x<3\\x\geq3\end{cases}

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

Solve each original inequality separately, reversing direction only when dividing by a negative number.
Take their common part: $\varnothing$ .
Check every included boundary in the original inequalities.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

Find the integer solutions of $2<x<4$ . Explain why its real solution set is not a single point.

\begin{cases}x>2\\x<4\end{cases}

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

The only integer strictly between 2 and 4 is 3.
Real numbers such as 2.5 and 3.5 also satisfy the inequalities, so the real interval contains more than one point.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

There are 29 toys and x children. Giving 3 each leaves at least 8 toys; giving 5 each would be short by 1 to 4 toys inclusive. Find the positive integer child count.

“And” means the conditions hold at the same time.
Mark each restriction separately before taking their overlap.
Preserve open/closed endpoints and impose an integer restriction only when the situation requires it.

The three restrictions are $29-3x\geq8$ , $5x-29\geq1$ , $5x-29\leq4$ .
They give $x\leq7$ and $6\leq x\leq\frac{33}{5}$ .
The only positive integer is 6; it leaves 11 with three each and is short 1 with five each.

Check your work

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Preparation and assessment notes

Prerequisites

Solve a linear inequality, mark open/closed endpoints and distinguish real values from whole-item counts.

Look for in students’ work

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original 40-minute core consolidation after the mapped textbook lessons; not a replacement for every first-teaching period. Only the supplied text has been mapped.

大湾区初中数学三维课堂（七年级下册）正文.pdf

Year 7, lower volume · text pp. 79–88 · textbook lessons 40–44

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Inequality systems: overlap and feasible choices

Year 7, lower volume · text pp. 79–88 · textbook lessons 40–44 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

A club pays a fixed 5 dollars plus 2 dollars per visit, has at most 27 dollars and needs at least 3 visits. Let x be the integer visit count. Form the restrictions and find all feasible counts.

Guided attempt2

Solve the system of inequalities; require every condition to hold.

\begin{cases}x>-1\\x\leq4\end{cases}

The first condition keeps values to the ____ of −1; that endpoint is open.
The second keeps values at or to the ____ of 4; that endpoint is filled.
Complete the overlap $-1\ \square\ x\ \square\ 4$ and draw it.

Use the stated conditions and the labelled scales.

Prerequisite3

“x is at most 7” means which inequality?

$x\geq7$
$x<7$
$x\leq7$

Prerequisite4

Is 2 included in $2<x\leq5$ ?

Yes; both endpoints are always included.
No; the lower inequality is strict.

Inequality systems: overlap and feasible choices

Year 7, lower volume · text pp. 79–88 · textbook lessons 40–44 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Solve the system of inequalities; require every condition to hold.

\begin{cases}3x-2>4\\2x+1\leq11\end{cases}

Use the stated conditions and the labelled scales.

Independent practice6

Solve the system of inequalities; require every condition to hold.

\begin{cases}x>4\\x\leq2\end{cases}

Independent practice7

Notebooks cost 3 dollars each with one fixed 2-dollar wrapping fee. With 20 dollars, buy at least one notebook. Let x be the count; find all feasible counts.

Independent practice8

Solve the system of inequalities; require every condition to hold.

\begin{cases}x\geq3\\x\leq3\end{cases}

Exit question9

List the integer solutions of $\frac{5}{2}<x<6$ .

\begin{cases}x>5/2\\x<6\end{cases}

Exit question10

A costs 2n dollars; B costs 10+n dollars. For n=14 and budget 25 dollars, which is feasible?

Only B: 24 dollars; A costs 28.
Both, because B is cheaper.
Only A.

Inequality systems: overlap and feasible choices — Further practice

Year 7, lower volume · text pp. 79–88 · textbook lessons 40–44

Further practice11

Solve the system of inequalities; require every condition to hold.

\begin{cases}x\geq1\\x>-2\end{cases}

Further practice12

An item costs 80 dollars. Let x be its selling price. The profit must be at least 15% of cost. Form an inequality and find the minimum price.

Further practice13

A container has capacity 11 L, initially 3 L. Each full bottle adds 2 L. Let x be a nonnegative integer bottle count. Find all counts that fit.

Further practice14

Solve the system of inequalities; require every condition to hold.

\begin{cases}x<3\\x\geq3\end{cases}

Extension15

Find the integer solutions of $2<x<4$ . Explain why its real solution set is not a single point.

\begin{cases}x>2\\x<4\end{cases}

Extension16

There are 29 toys and x children. Giving 3 each leaves at least 8 toys; giving 5 each would be short by 1 to 4 toys inclusive. Find the positive integer child count.

Inequality systems: overlap and feasible choices — Teacher answers

Year 7, lower volume · text pp. 79–88 · textbook lessons 40–44

Use the worked and guided tasks selectively. The eight core paper tasks are prerequisites, independent tasks and exits; further practice and extensions are optional.

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Solve a linear inequality, mark open/closed endpoints and distinguish real values from whole-item counts.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

Inquiry and discussion: 18 minutes using tariffs: A=2n, B=10+n, minimum 8 visits, budget 25. Predict which plan is suitable at n=9 and n=14 (3 min). Calculate and record both price and feasibility, then solve each plan’s budget and minimum restrictions on paper (7 min). Hold all prices/minimum fixed and change only budget to 15; check whether either overlap remains (5 min). Explain why “cheaper” and “feasible” are different, with exact integer endpoints (3 min). The existing controls support these parameter values.

Worked example1

A club pays a fixed 5 dollars plus 2 dollars per visit, has at most 27 dollars and needs at least 3 visits. Let x be the integer visit count. Form the restrictions and find all feasible counts.

Reference answer

$3\leq x\leq11$ , $x\in\mathbb{Z}$ .

3\leq x\leq 11

Cost gives $2x+5\leq27$ , and need gives $x\geq3$ .
The cost limit is $x\leq11$ ; intersect to get $3\leq x\leq11$ .
Every integer from 3 through 11 is feasible; both extreme counts meet the budget.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Guided attempt2

Solve the system of inequalities; require every condition to hold.

\begin{cases}x>-1\\x\leq4\end{cases}

The first condition keeps values to the ____ of −1; that endpoint is open.
The second keeps values at or to the ____ of 4; that endpoint is filled.
Complete the overlap $-1\ \square\ x\ \square\ 4$ and draw it.

Use the stated conditions and the labelled scales.

Reference answer

$-1<x\leq 4$ .

-1<x\leq 4

Solve each original inequality separately, reversing direction only when dividing by a negative number.
Take their common part: $-1<x\leq 4$ .
Check every included boundary in the original inequalities.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the stated conditions and the labelled scales.

Prerequisite3

“x is at most 7” means which inequality?

$x\geq7$
$x<7$
$x\leq7$

Reference answer

$x\leq7$

At most allows 7 but excludes larger values.
Thus x≤7.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Prerequisite4

Is 2 included in $2<x\leq5$ ?

Yes; both endpoints are always included.
No; the lower inequality is strict.

Reference answer

No; the lower inequality is strict.

Substituting 2 gives 2<2, which is false.
Use an open lower endpoint.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice5

Solve the system of inequalities; require every condition to hold.

\begin{cases}3x-2>4\\2x+1\leq11\end{cases}

Use the stated conditions and the labelled scales.

Reference answer

$2<x\leq 5$ .

2<x\leq 5

The first gives $x>2$ ; the second gives $x\leq5$ .
Both hold exactly on $2<x\leq5$ .

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Use the stated conditions and the labelled scales.

Independent practice6

Solve the system of inequalities; require every condition to hold.

\begin{cases}x>4\\x\leq2\end{cases}

Reference answer

$\varnothing$ .

\varnothing

One condition requires x greater than 4, the other at most 2.
No real number meets both; the solution set is empty.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice7

Notebooks cost 3 dollars each with one fixed 2-dollar wrapping fee. With 20 dollars, buy at least one notebook. Let x be the count; find all feasible counts.

Reference answer

$1\leq x\leq6$ , $x\in\mathbb{Z}$ .

1\leq x\leq 6

The restrictions are $3x+2\leq20$ and $x\geq1$ .
Cost gives x≤6; integer counts are 1,2,3,4,5,6.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Independent practice8

Solve the system of inequalities; require every condition to hold.

\begin{cases}x\geq3\\x\leq3\end{cases}

Reference answer

$x=3$ .

x=3

The number is at least 3 and at most 3.
Only x=3 meets both; the endpoint is included in each condition.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question9

List the integer solutions of $\frac{5}{2}<x<6$ .

\begin{cases}x>5/2\\x<6\end{cases}

Reference answer

$3\leq x\leq5$ , $x\in\mathbb{Z}$ .

\frac{5}{2}<x<6

Solve each original inequality separately, reversing direction only when dividing by a negative number.
Take their common part: $\frac{5}{2}<x<6$ .
Apply the stated integer restriction and check the extreme allowed integers.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Exit question10

A costs 2n dollars; B costs 10+n dollars. For n=14 and budget 25 dollars, which is feasible?

Only B: 24 dollars; A costs 28.
Both, because B is cheaper.
Only A.

Reference answer

Only B: 24 dollars; A costs 28.

A costs 28, exceeding 25.
B costs 24, within the budget.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice11

Solve the system of inequalities; require every condition to hold.

\begin{cases}x\geq1\\x>-2\end{cases}

Reference answer

$x\geq1$ .

x\geq1

Solve each original inequality separately, reversing direction only when dividing by a negative number.
Take their common part: $x\geq1$ .
Check every included boundary in the original inequalities.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice12

An item costs 80 dollars. Let x be its selling price. The profit must be at least 15% of cost. Form an inequality and find the minimum price.

Reference answer

$x\geq92$ .

x\geq92

Profit divided by cost gives $\frac{x-80}{80}\geq\frac{15}{100}$ .
Multiply by positive 80: $x-80\geq12$ , so $x\geq92$ .
The minimum selling price is 92 dollars.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice13

A container has capacity 11 L, initially 3 L. Each full bottle adds 2 L. Let x be a nonnegative integer bottle count. Find all counts that fit.

Reference answer

$0\leq x\leq4$ , $x\in\mathbb{Z}$ .

0\leq x\leq 4

Use $2x+3\leq11$ and $x\geq0$ .
Thus 0≤x≤4; counts 0 through 4 fit.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Further practice14

Solve the system of inequalities; require every condition to hold.

\begin{cases}x<3\\x\geq3\end{cases}

Reference answer

$\varnothing$ .

\varnothing

Solve each original inequality separately, reversing direction only when dividing by a negative number.
Take their common part: $\varnothing$ .
Check every included boundary in the original inequalities.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension15

Find the integer solutions of $2<x<4$ . Explain why its real solution set is not a single point.

\begin{cases}x>2\\x<4\end{cases}

Reference answer

$x=3$ , $x\in\mathbb{Z}$ .

2<x<4

The only integer strictly between 2 and 4 is 3.
Real numbers such as 2.5 and 3.5 also satisfy the inequalities, so the real interval contains more than one point.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.

Extension16

There are 29 toys and x children. Giving 3 each leaves at least 8 toys; giving 5 each would be short by 1 to 4 toys inclusive. Find the positive integer child count.

Reference answer

$x=6$ , $x\in\mathbb{Z}$ .

6\leq x\leq \frac{33}{5}

The three restrictions are $29-3x\geq8$ , $5x-29\geq1$ , $5x-29\leq4$ .
They give $x\leq7$ and $6\leq x\leq\frac{33}{5}$ .
The only positive integer is 6; it leaves 11 with three each and is short 1 with five each.

Use the given conditions and show a justified intermediate step.
State and check the requested result accurately.