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S3 Maths | Linear Inequalities in One Unknown Lesson-Prep Guide | flipping signs, number line, 4 mistakes
Students carry the negative sign across and write that minus two x greater than six gives x greater than minus three, and they mix up open and solid circles. This lesson does three things: reduce the method to three rules, draw every answer on a number line, and make every word problem start with a defined unknown.
Objectives — Students can: (1) solve inequalities of the form ax + b < c or ax + b ≥ c by transposition and explain that transposition is adding or subtracting the same amount on both sides; (2) explain that multiplying or dividing by a negative reverses the sign, and prove it by substituting x = -4 into -2x > 6; (3) show answers on a number line with an open circle for < and > and a solid circle for ≤ and ≥; (4) translate at least, at most, no less than and no more than into the four inequality signs and form the inequality from a defined unknown; (5) round word-problem answers to whole units and justify the direction of rounding.
01 Prerequisites (3-minute check)
If any of these fails, fix it before the lesson starts.
| Skill | Quick check | Pass mark |
|---|---|---|
| Solving linear equations | What is the first step in solving 2x + 3 = 11? | Subtract 3 from both sides to get 2x = 8 |
| Negative numbers on a number line | Which is greater, -3 or -5? | -3, because it lies further right |
| Inequality signs | What is the difference between < and ≤? | < excludes the endpoint, ≤ includes it |
02 Lesson flow (45 minutes)
| Time | Stage | Tutor does | Student does |
|---|---|---|---|
| 0-5 min | Wrong answer first | Write “-2x > 6, so x > -3” and ask students to test x = -4 | Substitute and report that 8 is not greater than 6 |
| 5-14 min | Three rules | Write the three rules: same add or subtract keeps the sign; same multiply or divide by a positive keeps it; by a negative flips it | Copy the three rules |
| 14-24 min | Flipping practice | Model 3x – 5 ≤ 7 and -4x + 1 < 9, forcing an arrow at the flip in the second | Work two in pairs and check each other |
| 24-33 min | Number line | Model open and solid circles and the direction of the arrow | Draw three answers on a number line |
| 33-40 min | Word problems | Model at most and at least sentences: define x, choose the sign, then round | Form two inequalities |
| 40-45 min | Exit check | Collect three items, mark on the spot, name the act to reteach | Hand in three items |
03 Teaching script (read it out)
Scene 1: Act 1: an inequality is an equation plus one flip rule
“Solving an inequality uses exactly the same moves as solving an equation: add or subtract the same amount, multiply or divide both sides, and keep x alone on the left. There is one difference, and it sits in the third rule: multiplying or dividing both sides by a negative reverses the sign. Why? A negative number turns the number line around. Three is greater than one, but multiply both by minus one and minus three is less than minus one. So for minus two x greater than six, divide by minus two and flip the sign to get x less than minus three. The only reliable check is substitution: with x equal to minus four, minus two times minus four is eight, and eight is greater than six, so the answer is right. The expensive mistake is copying the equation habit and writing x greater than minus three, which fails the moment a value is substituted.”
Multiplying or dividing by a negative flips the sign; always substitute one value to check.
Scene 2: Act 2: a number line needs a circle, and the circle follows the equals part
“An inequality answer has two forms: the inequality itself, and a stretch of the number line. The circle rule is rigid. For less than or greater than, draw an open circle, because the endpoint is excluded. For less than or equal to, or greater than or equal to, draw a solid circle, because the endpoint is included. The arrow follows the sign: x greater than two points right from two, and x less than or equal to minus one points left from minus one. Many students get the direction right and the circle wrong, and lose the mark anyway, so always ask whether the equals part is there. For a compound inequality such as minus one less than two x plus three less than or equal to seven, work on all three parts: subtract three to get minus four less than two x less than or equal to four, then divide by two to get minus two less than x less than or equal to two. If you divide three parts by a negative, both signs flip together; flipping only one is an error.”
Solid circle for the equals part, open circle without it; compound inequalities move on all three parts.
Scene 3: Act 3: define x first, then read the four key phrases
“Word problems need no creativity, only two moves: define the unknown, then recognise the keyword. At least, no less than and minimum all mean greater than or equal to. At most, no more than and maximum all mean less than or equal to. More than means strictly greater, and fewer than means strictly less. After defining x, write the comparison as one plain sentence before converting it: buying x notebooks at fifteen dollars each with a total of no more than one hundred and twenty dollars becomes fifteen x less than or equal to one hundred and twenty, so x is at most eight. The last step is rounding. People, books and sheets come in whole numbers, so x less than or equal to 8.6 becomes x less than or equal to 8. But if the question asks for the minimum needed, x greater than or equal to 8.6 rounds up to 9. The rounding direction follows the question, not the sign.”
Define x, read the keyword, then round according to what the question asks for.
04 Question types and worked steps
| Type | Example | Steps | Answer |
|---|---|---|---|
| Basic inequality | Solve 3x – 5 ≤ 7 and show the answer on a number line. | (1) Add 5 to get 3x ≤ 12 (2) Divide by 3, a positive, so the sign stays (3) Draw a solid circle at 4 (4) Point the arrow left | x ≤ 4, solid circle at 4, arrow to the left |
| Negative coefficient | Solve -4x + 1 < 9. | (1) Subtract 1 to get -4x < 8 (2) Divide by -4 and flip to x > -2 (3) Test x = 0: 1 < 9 holds (4) Open circle at -2, arrow right | x > -2 |
| Compound inequality | Solve -1 < 2x + 3 ≤ 7. | (1) Subtract 3 from all three parts: -4 < 2x ≤ 4 (2) Divide all three by 2: -2 < x ≤ 2 (3) Open circle at -2, solid at 2 (4) Check both signs moved together | -2 < x ≤ 2 |
| Word problem | A chair costs 45 dollars and Ming has 300 dollars. What is the greatest number of chairs he can buy? | (1) Let x be the number of chairs (2) At most means ≤, so 45x ≤ 300 (3) Solve: x ≤ 6.67 (4) Chairs are whole, round down to 6 | At most 6 chairs |
Sanity check (teach this): Reasonableness checks: (1) Substitute one value back into the original inequality; if it fails, a step is wrong. (2) A missed flip always fails on a negative test value, which makes it the fastest self-check. (3) The circle must match the equals part of the sign. (4) Word-problem answers must fit the real unit, so 6.5 chairs means rounding was skipped. (5) In a compound inequality, all three parts move together; flipping only one sign is an error.
05 Independent practice (3 questions)
| Question | Answer | What to watch |
|---|---|---|
| Solve 5x + 2 > 17 and show it on a number line. | x > 3, open circle at 3, arrow right | Did they flip when they should not have (positive coefficient)? |
| Solve -3x ≤ 12. | x ≥ -4 | Was the sign flipped after dividing by a negative? |
| A copier prints 24 pages a minute. What is the least number of whole minutes needed for a 500-page report? | Let t be minutes: 24t ≥ 500, t ≥ 20.83, so at least 21 minutes | Least means ≥ and rounds up |
06 Common mistakes and what to say
| Mistake | Student writes | Say this |
|---|---|---|
| No flip after dividing by a negative | “-2x > 6, so x > -3” | Divide by the negative and reverse: x < -3. |
| Wrong circle on the number line | x < 5 drawn with a solid circle | No equals part means an open circle. |
| Wrong sign in a word problem | Writing 45x ≥ 300 for the greatest number of chairs | At most means ≤; write the sentence in words first. |
| Rounding against the question | x ≤ 6.67 answered as 7 chairs | Round according to whether the question asks for the greatest or the least. |
07 Homework (paper, section and question numbers)
- Inequality practice, questions 1-4 on this page — Substitute one value to check each answer and draw it on a number line
- Prep-guide master index — The same transposition and flip rules extend to quadratic inequalities
- Redo this lesson — Write two word problems of your own, one asking for the greatest and one for the least
08 Exit check (last 5 minutes)
| Question | Answer | Decision |
|---|---|---|
| How does solving an inequality differ from solving an equation? | The moves are the same, except that multiplying or dividing by a negative flips the sign | No flip means reteach Act 1 |
| How do you show x < 3 on a number line? | Open circle at 3 with the arrow pointing left | Wrong circle or direction means reteach Act 2 |
| Which sign does at most use? | Less than or equal to | Answering greater than or equal to means reteach Act 3 |
09 Tutor FAQ
Can I teach only positive coefficients?
No. The flip is the one trap in the whole unit, so a negative coefficient must appear in the same lesson and students should ask about the coefficient before they start.
Should a compound inequality be split into two?
Either way works, but moving all three parts together is faster. Insist that both signs are written at the same time so only one gets changed by accident.
Should rounding rules be memorised?
No. Read whether the question asks for the greatest or the least, then check the answer against the real unit before writing it down.
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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.