At a glance. One lesson of about 60 minutes. It assumes solving linear inequalities in one variable from Year 8 and graphing lines from Year 9. Students use the definition to justify the reversal, solve double inequalities, and describe the region of a system with solid and dashed boundaries.
Calculator: GDC optional. The lesson is by hand. The optional section shows GDC users how shading works and why testing a point still matters.
What students will learn
Know
- a > b if and only if a − b > 0.
- Adding keeps the direction; multiplying by a positive keeps it; multiplying by a negative reverses it.
- Solid boundaries for ≤ and ≥, dashed for < and >; the region of a system is where all are true.
Understand
- The reversal follows from the definition, it is not a rule to memorize.
- A test point decides the side.
- A strict inequality can have a bound that is never reached.
Do
- Solve linear and double inequalities.
- Draw and shade the region of a system of linear inequalities.
- Find the corners of a region and count its whole-number points.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving inequalities, finding corners, counting points (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Students record what happens to the ray as the sign flips, and say why.
- Criterion C. Communicating: the proof in worked example 2 and the double inequality in the faded example, with the reversal stated.
- Criterion D. Applying mathematics in real-life contexts: constraints from the hall in the hook and the points counted in question 5.
The one thing to do with the figure
Make the reversal visible before it is named. Set a positive coefficient and ask for the solutions of the inequality. Then press Negative and ask students to say what happened to the ray before anyone writes the sign. Use the test value to confirm: pick a point on each side and check which is true.
In the second figure, ask for a test point that is on a boundary and ask students to read the three yes or no answers: the boundary counts for the ≤ and ≥ lines.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: which side, and does a negative coefficient change it? Collect predictions. |
| 10 min | First figure; the definition and the reversal; the warning card. |
| 10 min | Second figure; worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, return to the proof in worked example 2. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | x < 4; −2; 0 = 0, every point on the line. |
|---|---|
| Hook | a + s ≤ 60 and s ≥ 2a. For 6x > 18, x > 3; for −6x > 18 the side changes: x < −3. |
| Worked examples | x ≤ −5; proof by the definition; region with corner (3, 1) not included. |
| Faded example | −2 ≤ x < 1. |
| 1. Reversing the inequality | B. A keeps the sign; C loses the minus and keeps the sign; D loses the minus. |
| 2. Smallest integer solution | −3. x > −4. |
| 3. A double inequality | 6. −1 to 4. |
| 4. The corner of a region | 5. The corner is (3, 5). |
| 5. Counting whole-number points | 20. 8 + 6 + 4 + 2. |
| 6. Area of a region | 48. ½ × 12 × 8. |
| 7. Bounding a difference | 1. x − y < 2. |
| 8. Reading a region | C. A fails at the origin; B is not strict; D is the same as A. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Kept the sign, or lost the minus on 9. | Test one value from each side in −6x > 54. |
| −4 or 5 in question 2 | Included a strict boundary, or divided by −4 and got +4. | Write x > −4 and list the integers above it. |
| 7 or 5 in question 3 | Included or left out an end. | Test the end values in the original double inequality. |
| 3 or 7 in question 4 | Gave x, or put x = 3 into 2x + 1 instead of 2x − 1. | Substitute x back into the line y = 2x − 1. |
| 16 or 9 in question 5 | Treated ≤ as <, or ≥ as >. | Count column by column and write the count for each x. |
| 96 or 24 in question 6 | Left out the half, or used the wrong height. | Mark the three corners on a sketch. |
| 2 or 7 in question 7 | Used an unreachable bound, or added instead of subtracting. | Write x < 3 and −y < −1, and add. |
| A, B or D in question 8 | Tested the wrong inequality, or ignored the dashed line. | Test the origin in each option. |
Questions worth asking
- Why does multiplying by a negative reverse the sign? Listen for: a positive number times a negative is negative, so a − b > 0 turns into a negative.
- Why test a point rather than guess the side? Listen for: the side depends on the coefficients, and a test is fast and sure.
- What does a dashed boundary say? Listen for: the points on the line are not solutions.
- Can a strict inequality have a largest value? Listen for: not always, as the values can get as close to the bound as you like.
Differentiation
A way in for anyone stuck
- Give a table: write the inequality, the boundary, a test point, true or false, shade or not.
- Colour each inequality in a different colour so the region is the part coloured by every pencil.
- Start with a number line for one inequality before using a plane.
Stretch
- Prove that if a > b > 0 then 1/a < 1/b.
- Describe the region |x| ≤ 2, |y| ≤ 3 as four inequalities and give its area.
Students learning in English as an additional language
“At least” means more than or equal to. “At most” and “no more than” mean less than or equal to. “Integer” means a whole number that may be positive, zero or negative, such as −7, 0 or 12. “Strict” means the end value is not included. “Region” is an area of the plane. Say “is greater than or equal to” for ≥ every time at first.
| Word | Plain meaning | Example |
|---|---|---|
| inequality | a statement that one expression is larger or smaller than another | x + 4 < 10 |
| strict inequality | one that uses < or > so the end value is not included | x > 0 |
| boundary line | the line where the two sides of an inequality meet | y = 3x in y ≤ 3x |
| region | the set of points that satisfy every inequality in a system | the triangle inside three lines |
| test point | a point not on the boundary that you try in the inequality | (0, 0) when it is not on the line |
| double inequality | two inequalities written together | 3 < x ≤ 8 |
Sentence starters for explaining aloud or in writing:
- I divided by a negative number, so …
- The boundary is dashed because …
- I tested the point … and it is …, so I shaded …
Two things not to say
“Just flip the sign.” Flipping needs a reason: the multiplier is negative.
“Shade below for less than.” Below depends on the form; a test point decides.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- SolveSolve 3x − 4 ≥ 11.
x ≥ 5. - TestIs (1, 2) in the region y > 2x?
No: 2 > 2 is false.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- SolveSolve 7 − 2x < 1.
x > 3. - Earlier work (lesson 4)Find the point where x + y = 6 and x − y = 2 meet.
(4, 2).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that a > b and c > d give a + c > b + d.
(a − b) and (c − d) are positive, so their sum (a + c) − (b + d) is positive. - ExtendHow many whole-number points satisfy x ≥ 0, y ≥ 0 and x + y ≤ 4?
15.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or button, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Squared paper and coloured pencils are enough to draw and shade regions. The With a GDC section is optional.