Quadratic inequalities

When is it above or below?

In 3 of 20 recent exams
Paper 1 or 2
2 to 4 marks

How it is set in the exam

Paper 1 (no calculator), 2–4 marks a part, as a stand-alone question or the interval step of a longer one. Typical (3 of 20 exam versions, 4 parts, all Paper 1 (no calculator)). On Paper 2 (calculator) it would sit inside a context ("for how long…", Example 3.2), but no such part has been set since 2021.

Spot it in the exam

Exam phrases to spot (3)

“solve $\ldots > \ldots$ ”
“find the values of $x$ for which…”
“for what values of $x$ …”

In disguise (6)

" $f(x) > g(x)$ " or "the graph of $f$ is above the graph of $g$ ": subtract first
"above / below a height of…", "more than / less than…" in a context
"for how long…": the answer is the difference between two roots
" $f$ is increasing" when $f'(x)$ is a quadratic (Where is f increasing or decreasing?)
two conditions at once, such as where $f$ is negative and increasing: the sign gives the interval between (or outside) the roots, and increasing or decreasing is one side of the vertex ( $x > h$ is the increasing side when $a > 0$ ). The answer is the overlap, e.g. from the vertex's x-coordinate to the larger root. $f'(x) > 0$ also scores, but the vertex is quicker when an earlier part has found it.
where the log of a quadratic is defined, e.g. $\ln\left(8 + 2x - x^2\right)$ : its argument must be positive (Logarithm graphs)

Looks like it, but isn't (2)

an inequality in $k$ that comes from $\Delta$ . That is Find k from the number of roots, where the inequality is one step.
a graph that is not a parabola, above or below a level, read off a given sketch or found on the GDC: Solve graphically (a sinusoidal model against a level: Above or below a level)

Worked example

Two functions, one interval

Paper 1 (no calculator)

Question

Let $f(x) = x^2 - 3x$ and $g(x) = x + 5$ . Solve $f(x) < g(x)$ . [4]

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Steps

Rearrange so one side is $0$ . Never divide by an expression containing $x$ .
Solve the equation $= 0$ : factorise on , or the GDC on .
Sketch the parabola through the roots, opening up if $a > 0$ and down if $a < 0$ .
Read off the region. Include the roots for $\le$ or $\ge$ . Write an "outside" answer as two inequalities.
In a context, check the domain (for example $t \ge 0$ ), and answer the question asked (an interval, or a length of time).

Common mistakes

Dividing by $x$
(or by anything that could be negative or zero). Rearrange and factorise instead.
Reading the wrong region
always sketch. With $a < 0$ the "between" and "outside" rules swap.
Dropping the endpoints for $\le$ / $\ge$
, or including them for $<$ / $>$ .

More mistakes (2)

Giving the roots instead of the answer
in "for how long" questions. The answer is one number: the difference.
Rounding before subtracting
$1.98 - 0.875 = 1.105$ is borderline, and $1.98 - 0.88 = 1.10$ is wrong. Use the unrounded values from the GDC (keep full-accuracy values).

Easy to confuse with

Next typeRational functions

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Easy to confuse with

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