Equations with a modulus Cambridge IGCSE Additional Mathematics revision
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In plain words
|something| = 7 means the something is 7 or −7. Every modulus equation comes down to that: two cases. The catch is that sometimes one of the answers is a fake, so it has to be checked.
5 things to know
- |ax + b| = c, where c ≥ 0: solve ax + b = c and ax + b = −c.
- |ax + b| = |cx + d|: solve ax + b = cx + d and ax + b = −(cx + d). Or square both sides.
- |ax + b| = cx + d: solve both cases, then check each answer in the original equation. An answer that makes cx + d negative must be rejected.
- |quadratic| = d: solve quadratic = d and quadratic = −d. There can be up to four solutions.
- A modulus can never equal a negative number, so |ax + b| = −3 has no solutions.
Worked example
Solve |x − 4| = 2x + 1.
- Case 1: x − 4 = 2x + 1 gives x = −5. Check: the right-hand side is 2(−5) + 1 = −9, which is negative. A modulus cannot be negative, so reject x = −5.
- Case 2: −(x − 4) = 2x + 1 gives 4 − x = 2x + 1, so x = 1. Check: |1 − 4| = 3 and 2 + 1 = 3. It works.
- The only solution is x = 1.
Worked example
Solve |3x + 1| = |x − 5|.
- Case 1: 3x + 1 = x − 5 gives 2x = −6, so x = −3.
- Case 2: 3x + 1 = −(x − 5) gives 4x = 4, so x = 1.
- Both sides are moduli, so both answers are valid: x = −3 or x = 1.
Tips and tricks
- When one side has no modulus, as in |x − 4| = 2x + 1, always check. One answer is often false.
- A sketch of both graphs shows how many solutions to expect.
It lands in your notebook with its questions as flashcards.
Equations with a modulus: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What are the solutions of |x| = 6?
Both have a size of 6.
What are the solutions of |x − 2| = 5?
x − 2 = 5 or x − 2 = −5.
How many solutions does |2x + 1| = −3 have?
A modulus is never negative.
How many solutions does |x² − 5| = 4 have?
x² = 9 and x² = 1 each give two.
What are the solutions of |3x| = 12?
3x = 12 or 3x = −12.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 10 marks. Write your answers on paper, then check them.
Equations with a modulus
Cambridge IGCSE Additional Mathematics 0606 · 10 marks · papermunch.org
Name ______________________________ Date ______________
Solve |4x + 2| = 10.[3]
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x = 2 or x = −3. 4x + 2 = 10 or 4x + 2 = −10.
Solve |2x − 1| = |x + 4|.[3]
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x = 5 or x = −1. 2x − 1 = x + 4, or 2x − 1 = −x − 4.
Solve |x² − 5| = 4.[4]
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x = 3, −3, 1 or −1. x² − 5 = 4 gives x² = 9. x² − 5 = −4 gives x² = 1.
Answers: Equations with a modulus
- 1. x = 2 or x = −3. 4x + 2 = 10 or 4x + 2 = −10.
- 2. x = 5 or x = −1. 2x − 1 = x + 4, or 2x − 1 = −x − 4.
- 3. x = 3, −3, 1 or −1. x² − 5 = 4 gives x² = 9. x² − 5 = −4 gives x² = 1.