TMUA Inequalities: Quadratics and Traps

Inequalities feel like a small extension of solving equations, and for linear cases they almost are. The trouble is that the TMUA rarely stops at the linear case. It pushes you into quadratic inequalities, inequalities with an unknown in the denominator, and reasoning questions where a single careless sign flip turns a correct method into a wrong answer. Worse, the wrong answers produced by the classic mistakes are exactly the ones the examiners list as tempting options. This guide covers what the TMUA actually tests, how to handle linear inequalities and the negative-multiply trap, the sketch-first method for quadratics that beats every algebraic shortcut, inequalities involving fractions, and the handful of traps that account for most dropped marks.

What the TMUA actually examines

Inequalities sit in the core AS-level pure content that fills most of Paper 1, and they show up in two distinct guises. Sometimes the inequality is the question: solve $x^2 - 5x + 6 \geq 0$, or find the range of values for which a fraction is positive. More often the inequality is the last step of a longer problem, the point where you have a quadratic in hand and need to state for which values it is positive or negative. Either way the underlying skill is the same, and it is one the TMUA syllabus topics lean on repeatedly.

What the examiners are really testing is whether you can keep the direction of an inequality under control while you manipulate it. An equation does not care which way you move things around, but an inequality does, and that asymmetry is where almost all the marks are won or lost. The questions are written so that the most natural slip, forgetting to flip the sign or trying to multiply through by something whose sign you do not know, lands you on a plausible but wrong option. Our guide to calculator-free techniques covers the wider habit of staying exact and unhurried, which matters a great deal here.

So the goal is not to memorise new rules. It is to make the direction of the inequality something you track automatically.

Linear inequalities and the negative-multiply trap

A linear inequality behaves almost exactly like a linear equation. You can add or subtract the same quantity from both sides freely, and you can multiply or divide both sides by any positive number, all without touching the inequality sign. So to solve $3x + 4 < 19$, subtract $4$ to get $3x < 15$, then divide by $3$ to get $x < 5$, and the sign never moves.

The single thing that changes is this: when you multiply or divide both sides by a negative number, the inequality sign reverses. To solve $-2x < 6$, dividing by $-2$ gives $x > -3$, not $x < -3$. The less-than has become a greater-than. The reason is easy to see with numbers: $3 < 5$ is true, but multiply both sides by $-1$ and you get $-3$ and $-5$, and $-3 > -5$, so the order has genuinely reversed. This is the most common error in the entire topic, and the examiners know it.

The safest way to avoid the trap is to refuse to multiply or divide by a negative at all. Instead of dividing $-2x < 6$ by $-2$, add $2x$ to both sides and subtract $6$, giving $-6 < 2x$, then divide by the positive $2$ to get $-3 < x$, which is the same as $x > -3$. By always arranging for the variable to end up with a positive coefficient, you never have to remember to flip anything, because you never create the situation that demands a flip. That small discipline removes a whole category of mistake.

One more point worth fixing early: a strict inequality ($<$ or $>$) excludes the boundary value, while $\leq$ or $\geq$ includes it. The TMUA writes options that differ only in whether the endpoint is included, so read the symbol carefully and carry it through every line.

Quadratic inequalities by sketching

This is where most candidates go wrong, and where a clean method pays off most. The instinct is to treat a quadratic inequality like a linear one and try to chase the sign across the equals line. Do not. The reliable method has three steps, and it never fails.

First, get everything onto one side so the other side is zero. To solve $x^2 - 5x + 6 \geq 0$, it is already in that form. If it were not, you would collect terms until it was.

Second, factorise and find the roots, which are simply the solutions of the corresponding equation. Here $x^2 - 5x + 6 = (x - 2)(x - 3)$, so the curve crosses the $x$-axis at $x = 2$ and $x = 3$.

Third, sketch the parabola and read off the regions. Because the coefficient of $x^2$ is positive, the parabola is a happy U shape that dips below the axis between its roots and sits above the axis outside them. The question asks where the expression is greater than or equal to zero, meaning on or above the axis, so the answer is $x \leq 2$ or $x \geq 3$. The endpoints are included because the inequality is non-strict. Had the question asked for $x^2 - 5x + 6 < 0$ instead, the answer would be the inside region, $2 < x < 3$, with the endpoints excluded.

The sketch removes all guesswork about which way round the answer goes. A positive $x^2$ coefficient gives a U that is negative between the roots; a negative coefficient gives an n shape that is positive between the roots. Knowing the orientation of the curve is exactly the kind of thing covered in graphs and transformations, and a two-second sketch settles in an instant what an algebraic argument can muddle. Train yourself to draw the rough parabola every single time, even when you think you can see the answer, because the sketch is faster than second-guessing.

Try one

Reading about a method only gets you so far. Here is a past-paper question that turns on exactly this kind of reasoning. Give it a real attempt against the clock before you reveal the solution, and watch how the inequality work is the decisive step rather than an afterthought:

If that felt awkward, it is usually because the sketch was skipped or a sign was lost in passing, both of which are fixable with a more disciplined method.

Inequalities with fractions

An inequality with the variable in the denominator, like $\frac{1}{x} < 2$, hides a real trap, and it is one the TMUA likes. The tempting move is to multiply both sides by $x$ to clear the fraction. The problem is that you do not know the sign of $x$, so you do not know whether to flip the inequality. If $x$ is positive you keep the sign; if $x$ is negative you must reverse it. Multiplying blindly throws away half the answer.

There are two safe ways through. The cleaner one is to multiply by the square of the denominator, which is always positive and so never flips the sign. For $\frac{1}{x} < 2$ you multiply both sides by $x^2$ to get $x < 2x^2$, then rearrange to $2x^2 - x > 0$, factorise as $x(2x - 1) > 0$, and sketch. The roots are $0$ and $\frac{1}{2}$, the parabola opens upwards, so the expression is positive outside the roots, giving $x < 0$ or $x > \frac{1}{2}$. Notice that the awkward negative branch, $x < 0$, falls out automatically, which is precisely the part a careless multiply would have lost.

The alternative is to consider the sign of the denominator in cases, solving once for $x > 0$ and once for $x < 0$ and combining. That works, but it is slower and easier to fumble under time pressure, so the multiply-by-the-square method is usually better on the day. Either way the lesson is the same: never multiply an inequality by a quantity whose sign you do not know, because that sign controls the direction of the answer.

Common traps to avoid

A small set of errors accounts for most dropped marks here, and the answer options are written to reward each of them.

• Forgetting to flip the sign. Multiplying or dividing by a negative without reversing the inequality. Arrange for a positive coefficient instead and the problem disappears.
• Treating a quadratic inequality like a linear one. Trying to chase the sign across the line rather than collecting to one side, factorising, and sketching. The sketch is not optional, it is the method.
• Multiplying by an unknown sign. Clearing a fraction by multiplying through by $x$ when you do not know whether $x$ is positive or negative. Multiply by $x^2$ or split into cases.
• Losing the endpoint. Confusing strict and non-strict inequalities, so $\leq$ becomes $<$ or the boundary value is wrongly included or excluded. Track the symbol through every line.
• Stating the wrong region. Reading off the inside of the parabola when you wanted the outside, or vice versa. The orientation of the sketch tells you which, so always draw it.

Most of these are not failures of algebra, they are failures of bookkeeping, and the TMUA is hard largely because that bookkeeping has to hold up at speed. A consistent sketch-first method is the cheapest insurance against the lot.

How to drill it

You make this reliable the same way you make anything reliable: short, frequent, mixed reps rather than one marathon session.

Begin by getting the core moves cold. Drill linear inequalities where the variable starts with a negative coefficient until rearranging to a positive coefficient is automatic, so the flip rule never even comes up. Then drill quadratic inequalities until the three-step routine, collect to one side, factorise, sketch, runs without conscious thought, and force yourself to draw the parabola every time even when the answer seems obvious.

Then mix it into everything else. Because inequalities are so often the final step of a larger problem, you rarely need dedicated inequality sessions. Whenever you practise quadratics, graphs or coordinate geometry and a question ends in a range of values, treat that ending as an inequality rep and apply the full method. That folds the practice into work you were doing anyway, the most efficient form there is.

Finally, do it under time pressure on real questions. Decontextualised drills build the reflex, but only timed, exam-style questions teach you to hold the inequality direction steady when the clock is running and the sign trap is buried three steps deep. A structured plan that builds this in is laid out in how to prepare for the TMUA. Get the method automatic early and a whole class of question stops being a risk.

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