PAPER · REASONING · PRACTICE

Linear inequalities: direction and endpoints

Keep equivalent conditions, reverse for negative division and show the complete solution set.

40 minutes · Year 7, lower volume · text pp. 71–78 · textbook lessons 36–39

Textbook reference: Year 7 lower · Chapter 9 Inequalities and systems of inequalities · text lessons 36–39

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 with number-line cards: start from −3<1 and predict what adding 5, multiplying by 2, or multiplying by −2 does to the order (3 min). Place the resulting pairs on a common number line and compare which operation reverses the positions (6 min). Solve 3x>6 and −3x>6 separately, then test x=3 and x=−3 in both original inequalities (6 min). Explain why subtraction alone does not reverse the sign and show the open endpoint required by a strict inequality (3 min).

Organise the key idea

Keep equivalent conditions, reverse for negative division and show the complete solution set.

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

Solve the inequality.

3-2x\geq7

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.

Collect variable terms on the left and constants on the right: $-2x\geq4$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x\leq-2$ .

Check your work

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

Guided attempt2

Solve the inequality.

2(x-1)<x+5

Expand $2(x-1)<x+5$ to $2x-\square<x+5$ .
Subtract x and add 2 on both sides: $x<5+\square$ .
Thus $x<\square$ ; decide whether the boundary is included.

Multiply 2 by both terms inside the bracket.
Subtract x from both sides, then add 2 to both sides.
The remaining x coefficient is positive; keep the strict direction and an open endpoint.

Collect variable terms on the left and constants on the right: $1x<7$ .
Divide by the positive coefficient; the direction stays the same.
The solution set is $x<7$ .

Check your work

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

Prerequisite3

Does x=3 satisfy $2x+1>8$ ?

No; 7 is not greater than 8.
Yes; all positive x work.

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.

Substitute 3 into the left side: 2×3+1=7.
7>8 is false.

Check your work

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

Prerequisite4

Start with −3<1. Multiplying both sides by −2 gives which true inequality?

$6<-2$
$6>-2$

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.

Products are 6 and −2.
A negative multiplier reverses the direction.

Check your work

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

Independent practice5

Solve the inequality and show its solution on the blank number line.

-3x>6

Use the stated conditions and the labelled scales.

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.

Collect variable terms on the left and constants on the right: $-3x>6$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x<-2$ .

Check your work

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

Independent practice6

Solve the inequality.

5x-4\leq11

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.

Collect variable terms on the left and constants on the right: $5x\leq15$ .
Divide by the positive coefficient; the direction stays the same.
The solution set is $x\leq3$ .

Check your work

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

Independent practice7

Solve the inequality.

\frac{x+1}{2}>\frac{x-2}{3}

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.

Collect variable terms on the left and constants on the right: $\frac{1}{6}x>\frac{-7}{6}$ .
Divide by the positive coefficient; the direction stays the same.
The solution set is $x>-7$ .

Check your work

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

Independent practice8

Solve the inequality.

4-2(x+1)\geq x-7

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.

Collect variable terms on the left and constants on the right: $-3x\geq-9$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x\leq3$ .

Check your work

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

Exit question9

Solve the inequality.

-x\leq4

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.

Collect variable terms on the left and constants on the right: $-1x\leq4$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x\geq-4$ .

Check your work

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

Exit question10

For $x\leq3$ , which endpoint marking is correct?

A filled point at 3, extending left.
An open point at 3, extending left.
A filled point at 3, extending right.

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.

Equality is allowed, so include 3.
Smaller numbers lie to the left.

Check your work

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

Further practice11

Solve the inequality.

2x+3<7

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.

Collect variable terms on the left and constants on the right: $2x<4$ .
Divide by the positive coefficient; the direction stays the same.
The solution set is $x<2$ .

Check your work

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

Further practice12

Solve the inequality.

3x+2\geq5x-6

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.

Collect variable terms on the left and constants on the right: $-2x\geq-8$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x\leq4$ .

Check your work

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

Further practice13

Solve the inequality.

-4(x-2)>12

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.

Collect variable terms on the left and constants on the right: $-4x>4$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x<-1$ .

Check your work

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

Further practice14

Read the given number line and write its solution set. The filled endpoint is included.

Use the stated conditions and the labelled scales.

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.

The marked region begins at 2 and extends right.
The filled endpoint includes equality: x≥2.

Check your work

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

Extension15

Solve the inequality.

2x+1<2x+3

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.

Collect variable terms on the left and constants on the right: $0x<2$ .
The resulting constant statement is always true.
The solution set is $x\in\mathbb{R}$ .

Check your work

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

Extension16

Solve the inequality.

2x+5\leq2x+1

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.

Collect variable terms on the left and constants on the right: $0x\leq-4$ .
The resulting constant statement is false.
The solution set is $\varnothing$ .

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

Compare signed numbers, solve simple equations and recognise strict versus non-strict inequality symbols.

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. 71–78 · textbook lessons 36–39

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

Linear inequalities: direction and endpoints

Year 7, lower volume · text pp. 71–78 · textbook lessons 36–39 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Solve the inequality.

3-2x\geq7

Guided attempt2

Solve the inequality.

2(x-1)<x+5

Expand $2(x-1)<x+5$ to $2x-\square<x+5$ .
Subtract x and add 2 on both sides: $x<5+\square$ .
Thus $x<\square$ ; decide whether the boundary is included.

Prerequisite3

Does x=3 satisfy $2x+1>8$ ?

No; 7 is not greater than 8.
Yes; all positive x work.

Prerequisite4

Start with −3<1. Multiplying both sides by −2 gives which true inequality?

$6<-2$
$6>-2$

Linear inequalities: direction and endpoints

Year 7, lower volume · text pp. 71–78 · textbook lessons 36–39 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Solve the inequality and show its solution on the blank number line.

-3x>6

Use the stated conditions and the labelled scales.

Independent practice6

Solve the inequality.

5x-4\leq11

Independent practice7

Solve the inequality.

\frac{x+1}{2}>\frac{x-2}{3}

Independent practice8

Solve the inequality.

4-2(x+1)\geq x-7

Exit question9

Solve the inequality.

-x\leq4

Exit question10

For $x\leq3$ , which endpoint marking is correct?

A filled point at 3, extending left.
An open point at 3, extending left.
A filled point at 3, extending right.

Linear inequalities: direction and endpoints — Further practice

Year 7, lower volume · text pp. 71–78 · textbook lessons 36–39

Further practice11

Solve the inequality.

2x+3<7

Further practice12

Solve the inequality.

3x+2\geq5x-6

Further practice13

Solve the inequality.

-4(x-2)>12

Further practice14

Read the given number line and write its solution set. The filled endpoint is included.

Use the stated conditions and the labelled scales.

Extension15

Solve the inequality.

2x+1<2x+3

Extension16

Solve the inequality.

2x+5\leq2x+1

Linear inequalities: direction and endpoints — Teacher answers

Year 7, lower volume · text pp. 71–78 · textbook lessons 36–39

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: Compare signed numbers, solve simple equations and recognise strict versus non-strict inequality symbols.

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

Inquiry and discussion: 18 minutes with number-line cards: start from −3<1 and predict what adding 5, multiplying by 2, or multiplying by −2 does to the order (3 min). Place the resulting pairs on a common number line and compare which operation reverses the positions (6 min). Solve 3x>6 and −3x>6 separately, then test x=3 and x=−3 in both original inequalities (6 min). Explain why subtraction alone does not reverse the sign and show the open endpoint required by a strict inequality (3 min).

Worked example1

Solve the inequality.

3-2x\geq7

Reference answer

$x\leq-2$ .

x\leq-2

Collect variable terms on the left and constants on the right: $-2x\geq4$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x\leq-2$ .

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

Guided attempt2

Solve the inequality.

2(x-1)<x+5

Expand $2(x-1)<x+5$ to $2x-\square<x+5$ .
Subtract x and add 2 on both sides: $x<5+\square$ .
Thus $x<\square$ ; decide whether the boundary is included.

Reference answer

$x<7$ .

x<7

Collect variable terms on the left and constants on the right: $1x<7$ .
Divide by the positive coefficient; the direction stays the same.
The solution set is $x<7$ .

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

Prerequisite3

Does x=3 satisfy $2x+1>8$ ?

No; 7 is not greater than 8.
Yes; all positive x work.

Reference answer

No; 7 is not greater than 8.

Substitute 3 into the left side: 2×3+1=7.
7>8 is false.

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

Prerequisite4

Start with −3<1. Multiplying both sides by −2 gives which true inequality?

$6<-2$
$6>-2$

Reference answer

$6>-2$

Products are 6 and −2.
A negative multiplier reverses the direction.

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

Independent practice5

Solve the inequality and show its solution on the blank number line.

-3x>6

Use the stated conditions and the labelled scales.

Reference answer

$x<-2$ .

x<-2

Collect variable terms on the left and constants on the right: $-3x>6$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x<-2$ .

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 inequality.

5x-4\leq11

Reference answer

$x\leq3$ .

x\leq3

Collect variable terms on the left and constants on the right: $5x\leq15$ .
Divide by the positive coefficient; the direction stays the same.
The solution set is $x\leq3$ .

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

Independent practice7

Solve the inequality.

\frac{x+1}{2}>\frac{x-2}{3}

Reference answer

$x>-7$ .

x>-7

Collect variable terms on the left and constants on the right: $\frac{1}{6}x>\frac{-7}{6}$ .
Divide by the positive coefficient; the direction stays the same.
The solution set is $x>-7$ .

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

Independent practice8

Solve the inequality.

4-2(x+1)\geq x-7

Reference answer

$x\leq3$ .

x\leq3

Collect variable terms on the left and constants on the right: $-3x\geq-9$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x\leq3$ .

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

Exit question9

Solve the inequality.

-x\leq4

Reference answer

$x\geq-4$ .

x\geq-4

Collect variable terms on the left and constants on the right: $-1x\leq4$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x\geq-4$ .

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

Exit question10

For $x\leq3$ , which endpoint marking is correct?

A filled point at 3, extending left.
An open point at 3, extending left.
A filled point at 3, extending right.

Reference answer

A filled point at 3, extending left.

Equality is allowed, so include 3.
Smaller numbers lie to the left.

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

Further practice11

Solve the inequality.

2x+3<7

Reference answer

$x<2$ .

x<2

Collect variable terms on the left and constants on the right: $2x<4$ .
Divide by the positive coefficient; the direction stays the same.
The solution set is $x<2$ .

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

Further practice12

Solve the inequality.

3x+2\geq5x-6

Reference answer

$x\leq4$ .

x\leq4

Collect variable terms on the left and constants on the right: $-2x\geq-8$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x\leq4$ .

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

Further practice13

Solve the inequality.

-4(x-2)>12

Reference answer

$x<-1$ .

x<-1

Collect variable terms on the left and constants on the right: $-4x>4$ .
Divide by a negative coefficient and reverse the inequality.
The solution set is $x<-1$ .

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

Further practice14

Read the given number line and write its solution set. The filled endpoint is included.

Use the stated conditions and the labelled scales.

Reference answer

$x\geq2$

x\geq2

The marked region begins at 2 and extends right.
The filled endpoint includes equality: x≥2.

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.

Extension15

Solve the inequality.

2x+1<2x+3

Reference answer

$x\in\mathbb{R}$ .

x\in\mathbb{R}

Collect variable terms on the left and constants on the right: $0x<2$ .
The resulting constant statement is always true.
The solution set is $x\in\mathbb{R}$ .

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

Extension16

Solve the inequality.

2x+5\leq2x+1

Reference answer

$\varnothing$ .

\varnothing

Collect variable terms on the left and constants on the right: $0x\leq-4$ .
The resulting constant statement is false.
The solution set is $\varnothing$ .

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