Inequalities
Solve an inequality like an equation, doing the same to both sides, but when you multiply or divide by a negative, reverse the inequality sign. Show the answer on a number line: open circle for < and >, closed for ≤ and ≥. Check with a value.
You need more than 50 points to pass, and a ride needs riders at least 120 cm tall. Not one answer: a whole range of them.
An inequality compares two sides with , , or . Its solution is usually a whole range of values, such as , which you can show on a number line.
You solve it like an equation, doing the same to both sides, with one new rule. This lesson finds the rule, and why it works.
- Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
- Try: Try it on the board and compare the result with your prediction.
- Investigate: Work out why it happens, one step at a time.
- Modify: Change the problem on purpose and see what happens.
- Practise: A few on your own, to check you have it.
- Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
A negative in front
Pick a value from your answer and check it in .
When is −2x more than 6?
Where are all the points that work: left or right of ?
is on the line . Drag it to a point where is more than 6 (above the dashed line). Was your prediction right?
Use the board. This ticks when you get there.
Solve, show, check
Here is how to solve and show an inequality.
- Solve like an equation: do the same to both sides.
- Add 4 to both sides: the sign stays.
- Divide by 3 (positive): the sign stays.
- On a number line: an open circle at 5 (: 5 is not included; would be closed), shaded left.
- Divide by (negative): reverse the inequality sign. but : a negative flips the order.
- Check a value from the solution in the first inequality.
Now solve and show it on the board.
Need a hint?
3x − 4 < 11: add 4, then divide by 3.
Another hint
x < 5: open circle at 5 (5 itself is not included), shade left.
Read Mia's working. Her check is a clue.
Solve , then show the solution: move , choose the circle and the direction.
Use the board. This ticks when you get there.
Mia solves . Click the line where her working goes wrong.
One change at a time
Each task changes one thing.
Need a hint?
2x ≥ 4, so x ≥ 2: a closed circle (2 is included).
Need a hint?
Add 1: x/2 < 4.
Another hint
Multiply by 2 (positive): x < 8.
Solve and show it.
Use the board. This ticks when you get there.
Solve and show it.
Use the board. This ticks when you get there.
A range: how many integers have ?
Inequalities
Review: solve .
Write one with a negative
Swap with a partner: solve theirs, and check a value.
Need a hint?
Start from x ≥ −2 and multiply both sides by −3: the sign reverses.
Another hint
−3x ≤ 6. Check x = 0: 0 ≤ 6. ✓
Write an inequality with in it whose solution is . Solve it on paper, then show the solution on the line.
Use the board. This ticks when you get there.
For teachers
Students solve one-variable inequalities like equations, reversing the inequality sign when multiplying or dividing by a negative, show solutions on a number line with open or closed circles, and check graphically and by substitution (AC9M8A02, node 3). About 45 minutes.
Open the teacher guide: answers, solutions, slides and a worksheet.
Students will be able to
- I can solve an inequality, reversing the inequality sign when I multiply or divide by a negative.
- I can show a solution on a number line with the right circle (open or closed) and direction.
- I can check a solution by substituting a value, or with a graph.
Before class
Do the Investigate board yourself first. Students need Solving harder equations, and Lines and regions (Linear relations) for the graph.
The one idea: inequalities solve like equations, except that multiplying or dividing by a negative reverses the inequality sign, because it flips the order on the number line (2 < 5 but −2 > −5). The common mistake is keeping the inequality sign; it is planted in the Predict and in Mia's working, whose own check fails.
Students show answers on the number line board (move P, choose the circle and the direction), so the board marks the solution, not the way it is written.
Running the lesson
- Predict: take a show of hands, then have students test x = 0 in −2x > 6.
- Run: ask where all the working points are, and what that says about the solution.
- Investigate: model the worked example, then the number-line argument for reversing (2 < 5, −2 > −5).
- Modify: before 5 − x > 2, ask how to get x on its own when it has a minus sign.
- Make: collect students' inequalities and check one value from each.
When students get stuck
- A student keeps the sign after dividing by a negative (Mia's mistake). Ask: Pick a value from your answer and check it in the first inequality. Fix: If it fails, the sign should have reversed.
- A student chooses the wrong circle. Ask: Is the end number itself a solution? Fix: < and >: no, open circle. ≤ and ≥: yes, closed circle.
- A student avoids the negative by moving x across. Ask: Does that work too? Fix: Yes: −2x > 6 → 0 > 6 + 2x → −6 > 2x → −3 > x. Same answer, no reversing needed.
Stretch and support
Stretch: Solve −4 < 2x + 2 ≤ 10 (a double inequality: −3 < x ≤ 4) and show it on the line.
Support: Check every answer by testing one value inside and one outside the solution.
Where it leads
This completes AC9M8A02: graphing lines and regions, solving equations, solving inequalities.
Where it leads: inequalities with two variables (regions) and linear programming in later years.
What each step is for
- a negative in front: Committing to a solution of −2x > 6 shows whether they divide by −2 and keep the sign, the slip this lesson is built around.
- when is −2x more than 6?: Dragging P along y = −2x to where it is above y = 6 shows the solution lies to the left of −3: the sign turned round.
- solve, show, check: Adding, subtracting, and multiplying or dividing by a positive keep the inequality sign; multiplying or dividing by a negative reverses it, because it flips the order on the number line. Mia keeps the inequality sign after dividing by −2.
- one change at a time: One change at a time: a closed circle, a negative x term, a fraction, then a range between two values, and the edge case: why the inequality sign reverses.
- inequalities: Solving with and without reversal, reading a number line and the largest integer, mixed; the review keeps equations fresh.
- write one with a negative: Writing an inequality with a negative x term whose solution is x ≥ −2, then showing it, reverses the lesson: from the solution back to an inequality that needs the reversal.
- Solve an inequality, reversing the inequality sign when I multiply or divide by a negative.
- Show a solution on a number line with the right circle (open or closed) and direction.
- Check a solution by substituting a value, or with a graph.