Solving linear inequalities and number lines
Solving one- and two-step inequalities, and showing the solution set on a number line
About three lessons of 45 to 60 minutes
An equation has one answer. An inequality has a whole crowd of them.
Solving x + 3 = 10 gives exactly one number: x = 7. But solving x + 3 < 10 gives every number that makes the statement true: 6, 5, 0, -100, 6.999... an entire infinite crowd of solutions, all of them less than 7. A number line is the natural way to show that whole crowd at once, instead of trying to list infinitely many numbers.
The algebra of solving an inequality is almost identical to solving an equation: do the same operation to both sides, in the same order. One crucial rule differs: multiplying or dividing both sides by a NEGATIVE number flips the direction of the inequality sign.
- A rollercoaster needs riders taller than 120 cmheight > 120, an inequality with infinitely many solutions, not one exact height
- You have at most £50 to spendtotal cost ≤ 50, a whole range of affordable combinations, not one fixed amount
- Water is liquid between 0°C and 100°Ca compound (two-sided) inequality: 0 < temperature < 100
- Solving -2x < 8 gives x > -4, not x < -4dividing by the negative -2 flips the inequality sign, the one rule that is easy to forget
What students will be able to do
Students will solve one- and two-step linear inequalities in one variable, correctly flipping the inequality sign when multiplying or dividing by a negative number, and represent a solution set (including a compound, two-sided range) on a number line using open circles for strict inequalities and closed (filled) circles for 'or equal to' inequalities.
- I can solve a linear inequality by doing the same operation to both sides, in the same order I would use to solve an equation.
- I know that multiplying or dividing both sides of an inequality by a NEGATIVE number flips the direction of the inequality sign.
- I can draw an open circle for a strict inequality (< or >) and a closed (filled) circle for a non-strict inequality (≤ or ≥) on a number line.
- I can shade a number line in the correct direction: right for 'greater than', left for 'less than'.
- I can read a number line back into inequality notation, including a compound (two-sided) range.
Standards this unit teaches
- GCSE Algebra #22UK GCSE Mathematics (DfE, England)Solving linear inequalities and representing the solution set
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Algebra" section, "Solving equations and inequalities", item 22, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "solve linear inequalities in one or two variable(s), and quadratic inequalities in one variable; represent the solution set on a number line, using set notation and on a graph". In the rendered DfE source, "solve linear inequalities in one" and "represent the solution set on a number line" are underlined (Foundation-assessed), while "or two variable(s), and quadratic inequalities in one variable" and "using set notation and on a graph" are bold (Higher tier). This unit therefore solves and represents linear inequalities in one variable on a number line only; it excludes two-variable and quadratic inequalities, set notation, and graphed regions.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Inequality
- a mathematical statement that two expressions are NOT necessarily equal, using <, >, ≤ or ≥ instead of =
- Solution set
- every value that makes an inequality true; usually an infinite range of numbers, not a single value
- Strict inequality
- an inequality using < or > (the boundary value itself is NOT included in the solution set)
- Open circle
- the number-line symbol for a boundary value that is NOT included in the solution set (used for < and >)
- Closed (filled) circle
- the number-line symbol for a boundary value that IS included in the solution set (used for ≤ and ≥)
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Solving a linear inequality
ConcreteSolving an inequality uses the exact same moves as solving an equation: whatever you do to one side, do to the other, undoing operations in reverse order. The only new rule appears when you multiply or divide by a negative number.
Solve 3x + 5 < 20. Subtract 5 from both sides: 3x < 15. Divide both sides by 3 (positive, so the inequality sign stays the same): x < 5.
Now solve -2x + 3 > 11. Subtract 3 from both sides: -2x > 8. Divide both sides by -2. Because -2 is NEGATIVE, the inequality sign flips from > to <: x < -4.
Solve the inequality -2x + 3 > 11, and check the answer by substituting a value from the solution set back in.
- Subtract 3 from both sides: -2x > 8.
- Divide both sides by -2, flipping the sign because -2 is negative: x < -4.
- Check with x = -5 (which satisfies x < -4): -2(-5) + 3 = 10 + 3 = 13, and 13 > 11 is true.
- Check that a value OUTSIDE the solution set fails: x = -3 gives -2(-3) + 3 = 9, and 9 > 11 is false, as expected.
Answer: x < -4
- Why does dividing by a negative number flip the inequality sign, but dividing by a positive number does not?
- How would you check your answer to an inequality, since there is no single value to substitute back in?
2. Showing a solution set on a number line
PictorialA number line turns an inequality into a picture: a circle marks the boundary value, and shading shows every value in the solution set. The circle style tells you whether the boundary itself counts.
For x > 3: draw an OPEN circle at 3 (since 3 itself is not greater than 3, it is not included), and shade to the RIGHT, toward larger numbers.
For x ≤ -1: draw a CLOSED (filled) circle at -1 (since -1 satisfies 'less than or equal to -1'), and shade to the LEFT, toward smaller numbers.
Show the compound inequality -2 ≤ x < 5 on a number line, and explain the circle at each end.
- At x = -2: the inequality is ≤, so -2 IS included. Draw a CLOSED (filled) circle at -2.
- At x = 5: the inequality is <, so 5 is NOT included. Draw an OPEN circle at 5.
- Shade the whole region between -2 and 5, since every value strictly between them (and -2 itself) is a solution.
Answer: A filled circle at -2, an open circle at 5, shaded between them.
- Why does -2 ≤ x < 5 need TWO different circle styles, one at each end?
- Could a number line for a compound inequality ever have shading going in both directions, off both ends of the line?
Common misconceptions and how to address them
MisconceptionThe inequality sign flips whenever you see a negative number anywhere in the problem.
Why it happens: Students overgeneralise the flip rule to any negative number they spot, rather than to the specific act of MULTIPLYING or DIVIDING both sides by a negative.
How to address it: The flip rule applies ONLY at the moment you multiply or divide both sides by a negative number. Adding or subtracting a negative number (e.g. 'subtract -3', which means add 3) never flips the sign. Ask 'am I multiplying or dividing by a negative right now?' before every step.
MisconceptionAn open circle and a closed circle are just a stylistic choice, either can be used for any inequality.
Why it happens: Students treat the circle as decorative rather than as carrying real mathematical meaning about whether the boundary value is included.
How to address it: Ask directly: 'is the boundary value itself a solution?' Substitute it into the original inequality. For x ≥ 4, is 4 ≥ 4 true? Yes, so 4 is included, so the circle must be closed (filled).
Guided practice (with answers)
1. Solve 4x - 7 ≤ 9.
Answer: x ≤ 4, because 4x ≤ 16, then divide by the positive 4.
2. Solve -3x + 1 < 10.
Answer: x > -3, because -3x < 9, then divide by -3 and flip the sign to get x > -3.
3. Describe how x ≥ -2 would be shown on a number line.
Answer: A closed (filled) circle at -2, shaded to the right.
4. A number line shows an open circle at 6, shaded to the left. Write the inequality.
Answer: x < 6
5. Write the inequality shown by a closed circle at 0 and an open circle at 7, shaded between them.
Answer: 0 ≤ x < 7
Independent practice worksheets
Practise solving inequalities and moving between inequality notation and a number line, with computed, never-wrong answer keys.
Differentiation
- Solve every inequality as if it were an equation first (find the 'boundary value'), then decide the direction of the shading separately as a second step.
- Keep a written reminder ('flip when multiplying/dividing by a NEGATIVE') visible while practising, and require students to say out loud whether each step is a flip step before writing it.
- Physically act out a number line on the classroom floor: stand at the boundary value, then walk in the shaded direction to show which numbers are included.
- Solve an inequality with the variable on both sides (e.g. 5x - 2 > 2x + 7) before isolating x.
- Investigate what happens to a compound inequality like -3 < 2x + 1 ≤ 9 when solved for x (operating on all three parts at once).
- Ask students to write their OWN real-world scenario that an inequality models (a speed limit, a minimum height, a budget) and solve it.
Assessment: exit ticket
A three-question exit ticket sampling basic solving, the negative-coefficient flip, and number-line representation.
1. Solve 2x + 5 < 17.
Answer: x < 6, because 2x < 12, then divide by the positive 2.
2. Solve -4x + 2 ≥ 18.
Answer: x ≤ -4, because -4x ≥ 16, then divide by -4 and flip the sign to get x ≤ -4.
3. Describe how x ≤ 2 would be shown on a number line.
Answer: A closed (filled) circle at 2, shaded to the left.
Teacher notes and timings
- Rough timing: Lesson 1 solving inequalities including the flip rule (section 1), Lesson 2 number-line representation both directions (section 2), Lesson 3 compound inequalities and the exit ticket.
- Curriculum note: DfE GCSE Algebra item 22 is Foundation for LINEAR inequalities in one variable represented on a number line only; quadratic inequalities, set notation and 2-variable graphed regions are Higher-tier (bold) and deliberately not covered here.
- The numberLine figure's open-circle convention (a mark with a white `color`, giving a hollow appearance against the page background) is new to this unit; the closed-circle default (the figure's usual honey fill) needed no change.
- Present and print both work: use the Print button for a clean handout, or project the numberLine figure and build the shading live with the class, asking students to predict the circle style before you draw it.