Find the sign changes before doing the algebra
A minus sign outside a modulus reflects the output; it does not make the quantity inside the modulus negative. Write down where the inside is zero, then work on each side of that point. A quick substitution on each branch is a useful check.
- Mark every point where an expression inside a modulus is zero. These divide the number line into cases.
- Replace each modulus with the correct expression on that interval.
- Check the resulting branches meet at the breakpoints, then test the answers in the original equation.
Worked example
A reflection with a shift
Sketch y = 2 − |x − 1|. Find its maximum and its intersections with the axes.
The breakpoint is x = 1. For x ≤ 1, |x − 1| = 1 − x, so y = x + 1. For x ≥ 1, |x − 1| = x − 1, so y = 3 − x.
The two lines meet at (1, 2). The graph rises to this point and then falls, so its maximum is 2. Setting y = 0 gives (−1, 0) and (3, 0); setting x = 0 gives (0, 1).
You can also start from |x|, move it one unit right, reflect it in the horizontal axis, and move it two units up. Both routes must give the same sketch.
Your turn
Two modulus expressions
Solve |x − 1| + |x + 2| = 5.
Show a hint
The breakpoints are −2 and 1. Between them, think of the expression as the sum of the distances from x to the two endpoints.
Show the full solution
If x ≤ −2, the equation is (1 − x) + (−x − 2) = 5, giving x = −3. This belongs to the interval.
If −2 ≤ x ≤ 1, the sum is (1 − x) + (x + 2) = 3, so there is no solution here.
If x ≥ 1, the equation is (x − 1) + (x + 2) = 5, giving x = 2. Both candidates satisfy the original equation. The solutions are −3 and 2.
Before moving on, explain why there are three intervals and why the middle one gives a constant.
What to practise next
Practise a few graph transformations, then modulus equations. Return to a mixed paper once you can choose the cases without help.
- Physics & Maths Tutor: algebra and functions
Use the transformations-and-graphs and modulus question packs, with their matching model answers.
- Oxford MAT additional multiple-choice tests
Use these for mixed problem-solving after focused practice; they are historical papers.
The examples and explanations on this page are part of Arij Asad’s teaching guides. Linked papers and worksheets belong to their named creators and publishers. PMT’s topic collections include exam-board questions and other credited materials.