Equations, Inequalities and Graphs

This topic packages the syllabus’s graph-reading and modulus skills. None of the mathematics is deep; all of the marks depend on disciplined casework and sketches actually drawn. Students who “save time” by skipping the sketch pay for it in wrong regions.

Modulus equations: $|ax + b| = k$ and friends

$|ax + b| = k$ ($k > 0$) splits into two linear equations: $ax + b = k$ and $ax + b = -k$. Write the split explicitly, the "$= \pm$" line is the method mark:

$|3x - 2| = 7 \to 3x - 2 = 7$ or $3x - 2 = -7 \to$ $x = 3$ or $x = -\frac{5}{3}$

Two moduli $|2x - 1| = |x + 4|$: square both sides. $(2x-1)^2 = (x+4)^2 \to 3x^2 - 12x - 15 = 0 \to x^2 - 4x - 5 = 0 \to x = 5$, $x = -1$, or use the $\pm$ split; both are accepted, squaring is harder to fumble.

Modulus = expression $|x - 3| = 2x$: solve both cases, then check each answer, $x - 3 = 2x$ gives $x = -3$, but then $2x = -6 < 0$ while a modulus can’t be negative: reject, with a written reason. The rejection sentence carries a mark (the classic invalid-solution trap).

Modulus inequalities

For $k > 0$: $|ax + b| < k$ unpacks to the single sandwich $-k < ax + b < k$; $|ax + b| > k$ unpacks to the two-region answer $ax + b > k$ or $ax + b < -k$. If memory ever wavers, sketch $y = |ax + b|$ against $y = k$ and read the picture, the graph never lies, the memorised rule sometimes does. Answers must be written as proper intervals: "$1 < x < 4$" or ”$x < -2$ or $x > 5$”, never an impossible chained inequality.

Graphs of modulus functions

$y = |f(x)|$: draw $y = f(x)$, reflect every below-axis portion upward. Marks attach to labelled features, the vertex of $y = |2x - 5|$ at $\left(\frac{5}{2}, 0\right)$, the $y$-intercept at $(0, 5)$. For sketch commands, shape + labels is the entire game. Modulus graphs also solve equations graphically: the solutions of $|2x - 5| = x$ are the $x$-coordinates where $y = |2x - 5|$ meets $y = x$, drawing both and marking intersections can be the intended method when the question says “hence”.

Cubic equations and inequalities, graphically

Given (or after factorising) a cubic with roots at $x = -2, 1, 3$:

1. Sketch: positive $x^3$ coefficient $\to$ rises to the right; mark the three crossings.
2. Equation $p(x) = 0$: the roots, read directly.
3. Inequality $p(x) > 0$: read the regions where the curve is above the axis, here $-2 < x < 1$ or $x > 3$.

Repeated roots change the picture: a double root touches the axis (no sign change). With root behaviour drawn correctly, inequality regions read off without algebra. Cubic inequality answers are almost always a union of two intervals, a single-interval answer to a three-root cubic should trigger a re-check.

Worked exam-style question

Solve $|2x - 3| > 5$.

$|2x - 3| > 5 \to 2x - 3 > 5$ or $2x - 3 < -5$ (M, the two-case split) $\to x > 4$ or $x < -1$ (A, A)

Three marks, fifteen seconds, if the split line is written. The single most common script error is producing only the positive case and one solution.

Common mistakes in this topic

• One case solved, one forgotten, half the solutions, half the marks
• Solutions of modulus = expression left unchecked against the modulus’s non-negativity
• "$<$" and "$>$" unpackings swapped (sketch when unsure)
• Cubic inequality answered with roots instead of regions
• Sketches without labelled intercepts/vertices, shape alone doesn’t carry the B marks

Modulus work returns inside functions (graphs of |f(x)|) and the casework discipline transfers to trig equations, where multi-solution bookkeeping is the same skill. On Paper 1 these questions are pure technique, no calculator could help anyway.

If two-case logic keeps collapsing under exam pressure, it’s a drillable fix, free 1-hour trial class with Teacher Rig, booked on WhatsApp.