IGCSE Additional Mathematics: Equations, Inequalities and Graphs Practice Questions
A quadratic inequality is solved by finding the critical values and then deciding, from a sketch, whether the solution lies between them or outside them. A modulus inequality of the form less than a value produces a single interval.
Topic 3 of Cambridge IGCSE Additional Mathematics 0606 punishes algebraic manipulation without a sketch. Deciding between an inside interval and an outside pair is the whole question, and a rough parabola settles it in seconds. The questions below make that step explicit.
What you need to know for Equations, Inequalities and Graphs
- Critical valuesThe roots of the corresponding equation. They divide the number line into regions to be tested.
- Inside or outsideFor an upward parabola, the expression is negative between the roots and positive outside them. Sketching removes all guesswork.
- Modulus inequality, less thanThe modulus of an expression less than k means the expression lies between minus k and k, giving a single interval.
- Modulus inequality, greater thanThe modulus of an expression greater than k means the expression is greater than k or less than minus k, giving two separate regions.
- Modulus graphsThe graph of the modulus of a function reflects any part below the x-axis up above it, producing a sharp vertex at each root.
IGCSE Additional Mathematics Equations, Inequalities and Graphs questions and answers
4 exam-style questions written to the 0606 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Solve the inequality the modulus of (x minus 3) is less than 5.
Show the worked answer
- The modulus of an expression being less than 5 means the expression lies between minus 5 and 5.
- So minus 5 is less than x minus 3, and x minus 3 is less than 5.
- Add 3 throughout: minus 2 is less than x, and x is less than 8.
- The solution is a single interval, from minus 2 to 8, excluding both endpoints.
Solve the inequality x2 minus 5x + 6 is greater than 0.
Show the worked answer
- Find the critical values by solving x2 minus 5x + 6 = 0.
- Factorising gives (x minus 2)(x minus 3) = 0, so x = 2 or x = 3.
- The coefficient of x2 is positive, so the parabola opens upwards and lies above the x-axis outside the roots.
- The solution is therefore x is less than 2, or x is greater than 3.
Solve the inequality x2 minus 2x minus 8 is less than or equal to 0.
Show the worked answer
- Factorise: x2 minus 2x minus 8 = (x + 2)(x minus 4).
- The critical values are x = minus 2 and x = 4.
- The parabola opens upwards, so it lies below the x-axis between the roots.
- Since the inequality includes equality, the endpoints are included: minus 2 is less than or equal to x, which is less than or equal to 4.
Describe the graph of y equals the modulus of (2x minus 4), stating the coordinates of its vertex.
Show the worked answer
- The graph of y = 2x minus 4 is a straight line crossing the x-axis where 2x minus 4 = 0, so at x = 2.
- Taking the modulus reflects every part of the line that lies below the x-axis upwards.
- The result is a V shape with its vertex at the point (2, 0), where the original line met the axis.
- The right branch has gradient 2 and the left branch has gradient minus 2.
Common mistakes in this topic
- Choosing the wrong side of the critical values.
- Treating a modulus less than inequality as two separate regions.
- Using strict inequalities when the original includes equality.
- Solving an inequality by dividing by a variable, which can change the direction of the sign.
- Drawing a modulus graph as a smooth curve.
Exam tips
- Always sketch the parabola. It takes ten seconds and decides between inside and outside.
- Never divide an inequality by an expression containing x, since its sign is unknown.
- Less than gives one interval. Greater than gives two regions. Learn that pair once.
- Carry the equality through: less than or equal to in the question means the endpoints are included.
- For modulus graphs, find where the expression is zero first. That is the vertex.
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Equations, Inequalities and Graphs FAQs
How do I solve a quadratic inequality?
Rearrange so one side is zero, factorise to find the critical values, then sketch the parabola. For an upward parabola the expression is negative between the roots and positive outside them, which tells you which region satisfies the inequality.
What is the difference between a modulus less than and greater than inequality?
The modulus of an expression being less than k means the expression lies between minus k and k, giving one interval. The modulus being greater than k means the expression is above k or below minus k, giving two separate regions.
Why can't I divide an inequality by x?
The sign of x is unknown, and dividing by a negative reverses the inequality. Dividing by a variable can therefore produce a wrong answer or lose solutions. Rearrange to compare with zero and factorise instead.
What does the graph of a modulus function look like?
Any part of the original graph lying below the x-axis is reflected upwards. For a linear function this gives a V shape with a sharp vertex where the original crossed the axis, and the two branches have gradients equal in size but opposite in sign.
Continue through the IGCSE Additional Mathematics syllabus
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Written to the published Cambridge IGCSE Additional Mathematics (0606) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.