S3 Maths — Linear Inequalities and Inequality Systems: Flipping Signs and Counting Integer Solutions (Lesson Prep Guide)

2026-11-15

Tutor lesson-prep guides · Secondary Maths

Students rearrange equations fluently, then treat the inequality sign as decoration: they divide by a negative factor without flipping and cannot see that the answer is wrong. This lesson uses one move, substituting the boundary value, so students catch the error themselves.

Objectives — Students can: (1) solve a linear inequality and flip the inequality sign whenever they multiply or divide by a negative number; (2) write the solution set as an inequality and show it on a number line, using a solid dot when the endpoint is included; (3) solve a system of two inequalities, take the intersection and count the integer solutions inside it.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Solving linear equations	Solve 3x minus 5 equals 7	x = 4
Number lines	Which integers lie between minus 3 and 2?	Minus 2, minus 1, 0 and 1
Reading symbols	How do x greater than 4 and x not less than 4 differ?	The first excludes 4, the second includes 4

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-5 min	One wrong line	Write minus 2x greater than 8, divide by minus 2 to get x greater than minus 4, and ask if it is right	Substitute x = -10 to expose the error
5-15 min	Symbols and dots	Cover the four symbols and show solid against open dots	Draw two number lines themselves
15-29 min	Why the sign flips	Work minus 2x greater than 8 by adding 2x to both sides, then explain the flip	Explain why the sign flips
29-38 min	Intersection	Show x greater than 1 and x not greater than 5, overlap the two lines	Shade the overlap and count integers
38-42 min	Independent practice	Three questions on negative factors, a system and a best-buy problem; tour asking for the boundary check	Substitute the boundary every time
42-45 min	Exit check	Collect answers and ask when the sign flips	When multiplying or dividing by a negative

03 Teaching script (read it out)

Scene 1: Four symbols, four readings

Greater than, less than, not less than, not greater than. The open side of the symbol faces the larger number, which is worth saying once. Not less than 4 means greater than or equal to 4, drawn as a solid dot; greater than 4 excludes 4 and uses an open dot. That single dot decides whether the count of integer solutions is one more or one fewer. Ask each student to read all four symbols back.

Students distinguish solid from open dots

Scene 2: Multiply or divide by a negative and flip

For minus 2x greater than 8, dividing both sides by minus 2 flips the symbol, giving x less than minus 4. The plain reason: multiplying by a negative reverses the order, so the larger becomes smaller and the smaller becomes larger. When in doubt substitute: x = -10 makes the left side 20, and 20 really is greater than 8, so -10 must be inside the answer; writing x greater than minus 4 would exclude it and contradict itself. Ask students to explain the flip in their own words.

Students give the reason for flipping on a negative division

Scene 3: A system is an intersection, not a union

Two inequalities holding at once means the overlap of the two ranges, not everything covered by either. For x greater than 1 and x not greater than 5, the overlap is the stretch between 1 and 5, including 5. Draw both lines in different colours and the overlap is the answer; count integers by pointing at each dot, not in the head. The commonest failure is reading and as or, which turns the answer into the whole line.

Students explain the difference between and and or

04 Question types and worked steps

Type	Example	Steps	Answer
Solve an inequality	Solve 3x minus 5 is greater than 7	1) Add 5 to get 3x greater than 12. 2) Divide by 3, a positive factor, so the sign stays	x greater than 4
Negative factor	Solve minus 2x plus 1 is not greater than 9	1) Subtract 1 to get minus 2x not greater than 8. 2) Divide by minus 2 and flip	x is not less than minus 4
System	Solve x greater than 1 and x not greater than 5 and count integer solutions	1) Draw both on one number line. 2) Take the overlap. 3) Count integers one by one	Between 1 and 5 excluding 1 and including 5: four integers
Rounding down in context	Notebooks cost 8 dollars each and you have 50 dollars. What is the largest number you can buy?	1) Let n be the number, write 8n not greater than 50. 2) n not greater than 6.25. 3) Take the whole number below	Six notebooks

Sanity check (teach this): Three checks: first, look back for every multiplication or division by a negative and confirm the sign was flipped; second, substitute the boundary value into the original inequality, and if the two sides are merely equal while the symbol excludes equality, that point must drop out; third, make sure the answer is an inequality range rather than a single value, and when two inequalities combine, state clearly whether each endpoint is included. Numeric anchors: 3 times 4 minus 5 equals 7 exactly, so x = 4 fails and the answer is strictly greater; 8 times 6 is 48 and 8 times 7 is 56, so six notebooks fit the budget and seven do not.

05 Independent practice (3 questions)

Question	Answer	What to watch
Solve 5 minus 2x is not less than 11	x is not greater than minus 3	Check the sign was flipped after dividing by minus 2
x is not less than minus 1 and x is less than 3. How many integer solutions?	Four: minus 1, 0, 1 and 2	Check the negative end was not dropped
Notebooks cost 8 dollars each and you have 50 dollars. Largest number you can buy?	Six	Check 6.25 was not rounded up

06 Common mistakes and what to say

Mistake	Student writes	Say this
No flip after a negative division	minus 2x greater than 8 gives x greater than minus 4	Dividing by a negative flips the sign, so x is less than minus 4
Reading and as or	x greater than 1 and x not greater than 5 becomes the whole number line	Both must hold, so take only the overlap
Wrong dot	x greater than 4 drawn with a solid dot	Excluded endpoints get an open dot, included ones a solid dot
Rounding up in context	6.25 becomes seven notebooks	You cannot buy part of a notebook, so take six

07 Homework (paper, section and question numbers)

Solving simultaneous equations by elimination, question 2 — Two relations must hold at the same time
Quadratic formula and discriminant, question 3 — Same habit of testing values against the original statement
Polynomial operations, question 4 — Rearranging must be exact, or a correct direction still scores zero

08 Exit check (last 5 minutes)

Question	Answer	Decision
Solve 4x plus 3 is less than 19	x is less than 4	Answering x greater than 4 means rearranging lost the direction
Solve minus 3x is greater than 12	x is less than minus 4	Answering x greater than minus 4 means the flip was forgotten
x greater than 2 and x not greater than 4. Largest integer solution?	4	Answering 3 means not greater was read as less than

09 Tutor FAQ

Can not greater than be written as less than?

No, they differ by one point. Writing not greater than as less than is one of the commonest lost marks because the count drops by one.

Must systems be drawn?

Draw them while learning. Two lines overlapped show the intersection instantly; the shortcut of taking the narrower side comes later.

When do you round down in context?

Whenever the quantity cannot be split, such as notebooks or passengers, round down for a maximum and up for a minimum number of vehicles. Decide the direction before writing.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.