S3 Maths | Linear Inequalities in One Unknown Lesson-Prep Guide (for tutors) | 60 minutes from solving to integer solutions

2026-11-07

Tutor lesson-prep guides · Secondary Maths

Inequalities use exactly the same moves as equations, with one exception: multiplying or dividing by a negative number flips the sign. Nearly every student error traces back to that single step, and the rest come from never checking the answer.

Objectives — Students can: (1) solve a linear inequality using the same rearranging as an equation and reverse the sign correctly when multiplying or dividing by a negative; (2) show the solution on a number line with open circles and solid dots for less-than, at-most, greater-than and at-least; (3) list the integer solutions inside a given range and verify them by substitution.

01 Prerequisites (3-minute check)

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

Skill	Quick check	Pass mark
Solving linear equations	How do you solve 3x-5=7?	Add 5 to both sides, then divide by 3
Multiplying negatives	What is -6 divided by -2?	+3
Order on the number line	Which is larger, -3 or -1?	-1, further right is larger
Reading inequality signs	Does x at most 4 include 4?	Yes, use a solid dot

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-15 min	Equation and inequality side by side	Solve the same question twice, once with an equals sign and once with at-most, and see where the steps diverge	State that rearranging is identical and only the sign step differs
15-27 min	Moving and collecting terms	Show 5x+1 < 3x+9: subtract 3x, subtract 1, divide by 2	Four questions that never divide by a negative
27-37 min	Negative coefficients	Show -2x > 6 and 4-3x at least 13, writing “flip” beside the step each time	Four questions, marking the flip step
37-47 min	Number-line notation	Draw all four signs: open or solid, left or right	Four number lines with the endpoint labelled
47-55 min	Integer and compound solutions	Split -1 at most 2x+3 < 7 into two inequalities, intersect, then list integers	Three questions with integer lists
55-60 min	Exit check	Three quick oral questions	Independent answers

03 Teaching script (read it out)

Scene 1: Only multiplying or dividing by a negative flips the sign

The error pattern is consistent: students see a minus sign and flip, even in something like -3x+5 at most 7 where only adding and subtracting happen. Explain the mechanism instead of a slogan. Multiplying both sides by a negative reverses the order of the whole number line, so the direction must flip; adding or subtracting a negative just shifts everything, and order is preserved.
Prove it with numbers: 2 < 3, multiply both by -1 to get -2 and -3, and now -2 > -3. That single numeric example does more than any rhyme.

Students do: explain in their own words why multiplying by a negative flips the sign, using 2 < 3.

Scene 2: Always substitute a value back to check

An inequality has no single answer, so students cannot tell by inspection whether they are right. Build the habit: pick one number from the solution set and one from outside it, substitute both into the original inequality and see which holds.
Solving -2x > 6 gives x < -3; substituting x = -4 gives 8 > 6, true, while x = 0 gives 0 > 6, false. One true and one false substitution confirms the direction was not flipped by mistake, and it catches arithmetic slips as well.

Students do: after every solution, substitute one value inside and one outside the solution set.

Scene 3: Open circle versus solid dot is a mark, not decoration

At-most and at-least include the endpoint, so the dot is solid; less-than and greater-than exclude it, so the circle is open. Students draw a solid dot for an open interval and lose the mark even when the algebra is perfect.
Require three actions per number line: label the endpoint value, choose open or solid, draw the arrow. All three, every time, with the sign written above the endpoint for self-checking.

Students do: label endpoint, dot type and arrow direction on all four number lines.

04 Question types and worked steps

Type	Example	Steps	Answer
Basic rearranging	5x+1 < 3x+9	Move terms, collect, divide by the coefficient; no flip	x < 4
Dividing by a negative	-2x > 6	Divide both sides by -2 and change > to <	x < -3
Mixed with a negative coefficient	4-3x at least 13	Subtract 4 to get -3x at least 9, then divide by -3 and flip	x at most -3
Compound inequality	-1 at most 2x+3 < 7	Split into two inequalities, intersect, then list integers	-2 at most x < 2, integers -2, -1, 0, 1

Sanity check (teach this): Sanity checks: the sign must flip only when multiplying or dividing by a negative; verify the solution set by substituting one value inside and one outside; at-most and at-least take a solid dot while less-than and greater-than take an open circle; a compound inequality must satisfy both sides; when listing integers check each boundary, including endpoints for at-most and excluding them for less-than; fractional answers must be in simplest form.

05 Independent practice (3 questions)

Question	Answer	What to watch
Four questions without a negative divisor	Rearranging and collecting only	Check whether the sign is flipped for no reason
Four questions that must flip	Write “flip” beside that step	Check that flipping happens only on division by a negative
Three number-line and integer questions	Solve, draw the line, list the integers	Check dot type and boundaries

06 Common mistakes and what to say

Mistake	Student writes	Say this
Flipping at the sight of any minus sign	-3x+5 at most 7 answered with the sign reversed	That step only adds or subtracts, so nothing flips. The flip belongs to multiplying or dividing by a negative.
Forgetting to flip on division	-2x > 6 answered as x > -3	Substitute x = 0: 0 > 6 is false, so your solution set contains values that do not work.
Drawing the wrong dot	x at most 4 drawn with an open circle	The sign includes 4, so the dot is solid; an open circle says 4 is excluded.
Solving only one side of a compound inequality	-1 at most 2x+3 < 7 handled on the left only	The x in the middle must satisfy both sides, so split it and intersect.

07 Homework (paper, section and question numbers)

Secondary maths chapter index — All chapters for this subject
Lesson-prep guide index — Matching chapters in other subjects
Redo this chapter — Redo the four flip questions with two substitutions each

08 Exit check (last 5 minutes)

Question	Answer	Decision
When does the sign flip	When multiplying or dividing by a negative	Answering “whenever I see a minus” means returning to act three
Open or solid for x at most 4	Solid, 4 is included	Answering open means returning to act four
How to handle a compound inequality	Split it and intersect	Solving it in one go means returning to act five
How to check a solution	Substitute one value inside and one outside	Not checking means returning to act two

09 Tutor FAQ

What if arithmetic is weak and division keeps going wrong?

Let the student write the division as a fraction and cancel, with every step on the page instead of mental working.

Should inequalities be read aloud?

Yes: saying “x is at most 4” prevents the confusion between less-than and at-most, especially in word problems.

How should word problems start?

Define the unknown, write one inequality containing the key phrase such as at least or no more than, solve, then check the answer against the real situation.

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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.