At a glance. One lesson of about 60 minutes. It assumes lesson 7 and combining algebraic fractions. Students avoid multiplying by an expression of unknown sign, combine into one fraction, use a sign diagram with excluded values, solve a cubic, and read a solution from the graphs of two sides.
Calculator: GDC optional. A GDC is optional: the algebra is the assessed skill and the graph is a check. The GDC note shows how to find a crossing and warns that it gives regions, not end values.
What students will learn
Know
- A rational expression can change sign at a zero of the top or a zero of the bottom.
- A value that makes the bottom zero is never in the solution.
- Multiplying by an expression reverses the inequality where that expression is negative.
Understand
- Why multiplying by the denominator can lose solutions.
- Why combining into one fraction avoids that problem.
- Why a graph of both sides shows the same regions as a sign diagram.
Do
- Solve a rational inequality exactly with excluded values.
- Solve a cubic inequality by a sign diagram.
- Count integer solutions from a graph or sign diagram.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: combining and using sign diagrams (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say how the solution changes when the denominator's zero moves to the other side.
- Criterion C. Communicating: writing the solution set with the correct signs and exclusions.
- Criterion D. Applying mathematics in real-life contexts: cost per item in a batch (hook, cost per bag).
The one thing to do with the figure
Ask for the prediction first. Then set the zero of the numerator to 0 and move the denominator's zero left and right, and ask which end value is allowed and which is not.
Ask: “Why is there a gap in the curve at the dashed line, and why does the strip change colour there?” The fraction is not defined at that value and its sign flips across it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the cost per bag; collect predictions. |
| 12 min | Figure, key idea and warning card; solve 4 ÷ (x − 2) < 1 both ways. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C, ask what the fraction equals at x = −2 and whether it exists at x = 5. |
| 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 | −3 < x < 3; (1 − x) ÷ (x(x + 1)); x = 4. |
|---|---|
| Hook | For x > 0, (500 + 20x) ÷ x < 30 gives 500 + 20x < 30x, so x > 50: at least 51 bags in a batch. |
| Worked examples | −3 < x ≤ 0.5; x < 1 or x > 3; −2 < x < 0 or x > 2. |
| Faded example | x < −2. |
| 1. Reading a rational inequality | D. A is the positive region; B is one side of it; C includes both ends. |
| 2. Largest integer in a rational solution | 5. −2 < x < 6. |
| 3. Multiplying by a positive denominator | 2. x < 2.4. |
| 4. Excluded value and an included end | 10. −1 < x ≤ 4. |
| 5. Cubic inequality with a negative leading term | 6. −4, −3 and 0 to 3. |
| 6. Rearranging before solving | 6. x < 3 or x ≥ 6. |
| 7. Comparing two graphs | 3. −6, −5, 1. |
| 8. Solving by combining first | B. A multiplies without checking the sign; C and D are regions where the fraction is negative. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Gave the positive region, one side, or included both ends. | Ask them to test x = 0 in the original fraction. |
| 6 or −1 in question 2 | Included the zero of the numerator, or gave the smallest integer. | Ask: is 0 less than 0, and is the question about the largest or smallest? |
| 3 or 60 in question 3 | Rounded up, or multiplied instead of dividing. | Test the answer in the original inequality. |
| 9 or 6 in question 4 | Included the value that makes the denominator zero, or left out an end value. | Substitute x = −1 and x = 4 into the original inequality. |
| 7 or 3 in question 5 | Used the sign pattern of a cubic with a positive x3 term, or treated ≥ as >. | Test x = 4 and x = −4 in the original inequality. |
| 4 or 7 in question 6 | Reversed the inequality, or left out the end value 6. | Test x = 4, 5 and 6 in the original inequality. |
| 4 or 5 in question 7 | Included the crossing, or missed that x = 0 is excluded and reversed signs. | Draw the graphs and read where one is above the other. |
| A, C or D in question 8 | Multiplied without the sign check, or gave a region where the fraction is negative. | Test x = −4, x = 0 and x = 3 in the original inequality. |
Questions worth asking
- Why can we not multiply both sides by x − 3 when x is unknown? Listen for: it might be negative, which reverses the inequality.
- What is the difference between a zero of the numerator and a zero of the denominator? Listen for: one gives a value of 0, the other gives no value at all.
- How can you check the answer without algebra? Listen for: test a value from each region in the original inequality.
- What does the graph show that the sign diagram does not? Listen for: how fast the fraction changes and where the break is.
Differentiation
A way in for anyone stuck
- Give a pre-drawn sign diagram template with the factors listed in rows and the critical values marked.
- Provide a worked solution with one step missing for each of two similar questions.
- Give the frame: “Combined fraction: …. Critical values: …. Excluded: …. Solution: ….”
Stretch
- Solve (x − 1)(x + 3) ÷ (x − 2) ≥ 0.
- Show that 1 ÷ x < x has the solution −1 < x < 0 or x > 1.
Students learning in English as an additional language
“Rational” here means a fraction with algebra on the top and bottom. “Denominator” is the bottom of a fraction. “Excluded” means not allowed. “Asymptote” is a line that a graph gets close to but never touches.
| Word | Plain meaning | Example |
|---|---|---|
| rational expression | a fraction with algebra on the top and bottom | (x + 1) ÷ (x − 9) |
| denominator | the bottom of a fraction | the x − 9 in that fraction |
| excluded value | a value of x that makes an expression undefined | x = 9 for that fraction |
| asymptote | a line that a graph approaches but does not reach | the dashed line in the figure |
| sign diagram | a number line showing where an expression is positive or negative | plus, minus, plus |
Sentence starters for explaining aloud or in writing:
- The critical values are … because …
- I excluded … because the denominator would be …
- I checked by testing x = … in the original inequality.
Two things not to say
“Cross multiply.” Say instead: combine into one fraction and read the sign, unless the multiplier is known to be positive.
“The GDC will give the answer.” Say instead: the graph shows regions, and the inequality sign decides the ends.
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.
- FindSolve (x + 3) ÷ (x − 1) > 0.
x < −3 or x > 1. - FindFor x > 0, solve 20 ÷ x < 4.
x > 5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextThe cost per item is C = (300 + 15x) ÷ x. For x > 0 find when it is below 25.
300 < 10x, so x > 30. - Earlier work (Year 10)Solve x2 − 3x − 10 < 0.
−2 < x < 5.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSolve 3 ÷ (x + 1) ≤ 1.
x < −1 or x ≥ 2. - Working backwardsWrite a fraction that is positive exactly for x < 2 or x > 7 and undefined at x = 7.
(x − 2) ÷ (x − 7).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
A GDC is optional; the section is closed until a student opens it.