Deep Dive: Quadratic Inequalities Graphically

Provenance (2026-09-29). Read in full for this pass: research/veridian/WMA11-verified-facts.md (§1 spec table; §3 mark-scheme conventions; §4.1's complete "Quadratic inequalities interpreted/represented graphically" entry, re-read verbatim; §6; §7; the quadratic-inequalities-graphically.ts Lesson-audit-log entry, lines ~896–929); the live lesson file content/veridian/curriculum/wma11/quadratic-inequalities-graphically.ts (all 937 lines); signal-topic-brief-quadratic-inequalities-graphically.md (read in full); codex-formula-and-method-bank.md §1.1.5–§1.1.6 (read); codex-complete-worked-paper.md (grepped for "inequal", its Question 8(d) read); signal-frequency-and-boundaries.md (grepped, one row used); content/veridian/question-bank/wma11.ts (read in full — confirmed no quadratic-inequality item seeded there yet, so this document does not overlap it). Two fresh primary-source extractions were made for this document, neither already sitting in the facts bank in this form: (1) the actual Pearson specification PDF itself (international-a-level-maths-spec.pdf, Issue 3, April 2019) — page 14, "P1.3 Unit content" — rendered and read directly as a page image, pdftotext -layout and -raw cross-checked, for the Guidance column text of spec items 1.7–1.9, which the facts bank's own §1 does not transcribe (it carries only the "what students need to learn" column); (2) the real June 2023 WMA11 Unit 1 question paper, mark scheme and examiner report (Mathematics_Unit 1_QP/MS/ER_June_2023.pdf, Question Paper Log Number P72868RRA, Publications Code WMA11_01_2306_MS/_ER), independently re-extracted this pass — including a direct page-image render of the mark scheme, not merely the facts bank's own prior one-line spot-check note — for Question 1 in full. Every critical value, intersection and generosity-range endpoint below was independently recomputed by hand and checked against the real mark scheme's own stated answer before being written in, not copied blind. Not affiliated with or endorsed by Pearson Edexcel.

#CODEX Deep-Dive — Quadratic Inequalities, Interpreted and Represented Graphically (WMA11, spec 1.7–1.9)

#0. What this document is, and isn't

A single-topic reference for spec points 1.7, 1.8 and 1.9 only. It does not repeat the lesson file's three prequestions, its two teach blocks' full prose, its mechanism-block derivation verbatim, its diagram, its worked-chain, chain-drill, two marked-solutions, seven-item trap-taxonomy or six MCQs — those already exist, are pedagogically sequenced, and (per that lesson's own header) are deliberately built on numbers that differ from the real anchors. What follows instead is: the spec's own guidance-column wording, freshly verified against the primary PDF rather than inherited; the mechanism restated as compact reference rather than taught prose; every real, distinct sub-case this exact topic's evidence base actually supports, each worked on short numbers; the two richest real anchors worked on their own actual numbers (which the lesson deliberately avoids using); and a synthesis of every real WMA11 exam angle this topic has been tested from, including one anchor — June 2023 Q1 — that this document is the first place in this course to carry a full, independently-verified mark-scheme breakdown for.

Two labels are used throughout, matching this course's standing convention:

• [real] — a verified fact from a genuine Pearson question paper, mark scheme or examiner report, series and question number given.
• [VERIDIAN, illustrative] — a fresh, short, invented example built to demonstrate a mechanism, with no claim to resemble any specific real Pearson question.

No likelihood percentage or grade boundary is stated or implied anywhere below — the evidence available does not support either, and none is claimed.

#1. What spec 1.7–1.9 actually ask for — the guidance column, verbatim

The spec's "what students need to learn" column names three separate skills with three separate verbs, and it is worth keeping them apart rather than treating "quadratic inequalities" as one undifferentiated topic:

• 1.7 "Interpret linear and quadratic inequalities graphically" — reading meaning from a picture or an inequality.
• 1.8 "Represent linear and quadratic inequalities graphically" — producing the picture from an inequality, with "shading and use of dotted and solid line convention... required."
• 1.9 "Solutions of linear and quadratic inequalities" — actually solving one.

The guidance column (confirmed this pass directly against the primary spec PDF, p.14, "P1.3 Unit content", Issue 3 April 2019 — not previously transcribed into this course's facts bank, which carries only the left-hand "what students need to learn" column) gives each of these a worked template:

• 1.7's own three named example forms: "$ax+b>cx+d$, $px^2+qx+r\geq0$, $px^2+qx+r<ax+b$" — interpreting the third of these, verbatim, "as the range of $x$ for which the curve $y=px^2+qx+r$ is below the line with equation $y=ax+b$." This is the spec's own official framing of a quadratic-vs-line inequality as a height comparison between two graphs — exactly the reading the lesson's own lede independently arrives at, confirmed here as the spec's literal intent, not a course paraphrase of it.
• 1.7's own authorisation for reciprocal-type inequalities: verbatim, "including inequalities with brackets and fractions. These would be reducible to linear or quadratic inequalities, e.g. $\frac{a}{x}<b$ becomes $ax<bx^2$, $x\neq0$." This single clause is the spec's own named method for §3.4 below — multiplying a reciprocal inequality through by $x^2$ is not an exam-technique trick a mark scheme happens to reward; it is the spec's own stated reduction, in its own worked example.
• 1.8's own two named example forms: "$y>x+r$ and $y>ax^2+bx+c$" — confirming that representing an inequality graphically is explicitly expected to cover both the linear and the quadratic case, each drawn as a shaded half-plane or region bounded by a curve, with the dotted/solid convention stated as a hard requirement, not a stylistic suggestion.
• 1.9's own three example forms are the identical three inequalities named under 1.7 — confirming that 1.7 and 1.9 are the same underlying inequalities looked at from two angles: 1.7 asks for the graphical meaning, 1.9 asks for the actual solution set.

#2. The mechanism, stated as compact reference

Write $f(x)=a(x-p)(x-q)$ with $p<q$ the two distinct real roots (requiring $b^2-4ac>0$ strictly — spec 1.4, already covered as a prerequisite, and tightened here to its strict form: $p<q$ itself already excludes the repeated-root case). The product $(x-p)(x-q)$ is positive for $x<p$ (both factors negative), negative for $p<x<q$ (factors of opposite sign), positive again for $x>q$ (both positive) — a plain consequence of two linear expressions each changing sign only at their own zero. Multiplying by $a>0$ preserves every sign, giving the upward rule: $f(x)>0$ outside the roots, $f(x)<0$ between them. Multiplying by $a<0$ flips every sign, giving the exact reverse for a downward parabola. A single test point substituted into any one interval fixes that interval's sign directly and needs no memorised rule at all.

Strictness convention (spec 1.8's own stated requirement): a solid boundary/filled point means included ($\leq,\geq$); dashed/open means excluded ($<,>$). The original inequality's own strictness must be carried, unchanged, into the final written answer.

Two boundary cases spec 1.4's own three-way split implies but this topic's real evidence base has no anchor for [VERIDIAN, illustrative, for completeness only — no real WMA11 anchor applying a repeated or non-real root specifically to a quadratic inequality question has been found in this course's research]: if $\Delta=0$, $f(x)=a(x-p)^2$ never changes sign — for $a>0$, $x^2-6x+9\geq0$ (i.e. $(x-3)^2\geq0$) holds for every real $x$, while $x^2-6x+9<0$ has no solution at all, since a square is never negative. If $\Delta<0$, $f(x)$ again never changes sign across the whole real line — $x^2+2x+5>0$ (completing the square: $(x+1)^2+4>0$) holds for every real $x$, while $x^2+2x+5<0$ has no solution. Both are legitimate readings of the same "outside vs between the roots" logic taken to its boundary, not a separate rule — they simply have no roots (or one repeated root) to be "between" or "outside" at all.

#3. The real distinct sub-cases, with the evidence for each

#3.1 Foundational case — a single, already-arranged quadratic inequality with two distinct roots

The base case every other sub-case reduces to eventually: rearrange to "(quadratic) compared with 0" if not already there, solve for the critical values by any of the three mark-scheme-equal routes (factorisation, the formula, completing the square — verified, facts bank §3, "Method mark for solving a 3-term quadratic... by any of three named routes"), then read the region off the sign of $a$.

[VERIDIAN, illustrative] Solve $2x^2+3x-9\leq0$. Factorise: $(2x-3)(x+3)=2x^2+6x-3x-9=2x^2+3x-9$ ✓, critical values $x=\frac32$ and $x=-3$. $a=2>0$, so $\leq0$ is the between-the-roots case, non-strict: $-3\leq x\leq\frac32$. Check $x=0$ (between): $2(0)+3(0)-9=-9\leq0$ ✓.

#3.2 The inequality is not pre-arranged — rearrangement is the first, graded step — real anchor: June 2023, Q1 [real]

Every worked example in the live lesson file starts from an inequality already sitting with one side at zero, or explicitly reduces to that form as a stated first step. A real, low-tariff WMA11 question tests the plain, un-signposted version of that same first step directly, and — verified independently this pass, not merely inherited from the facts bank's own prior one-line spot-check — carries specific, quotable traps of its own.

The real question (4 marks, WMA11/01, Question Paper Log Number P72868RRA, Publications Code WMA11_01_2306_QP/_MS; the paper is dated "Tuesday 9 May 2023" on its own cover but is catalogued under Pearson's own _2306_ publication code as the June 2023 series, per this course's existing convention): "In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Solve the inequality $4x^2-3x+7\geq4x+9$." (Confirmed by direct visual inspection of the real question-paper page image this pass — the inequality is $\geq$, not $\leq$, which matters for everything that follows.)

Worked on the real numbers: $4x^2-3x+7\geq4x+9 \Rightarrow 4x^2-7x-2\geq0 \Rightarrow (4x+1)(x-2)\geq0$ (check: $4x\cdot x+4x\cdot(-2)+1\cdot x+1\cdot(-2)=4x^2-8x+x-2=4x^2-7x-2$ ✓), critical values $x=-\frac14,\ 2$. Since $a=4>0$ and the inequality is $\geq0$, this is the outside-the-roots case, non-strict: $x\leq-\frac14$ or $x\geq2$.

The real mark scheme, verbatim structure, re-extracted directly from the mark-scheme page image this pass: M1 "Gathers terms to one side and solves their 3TQ which is not $4x^2-3x+7=0$ to obtain at least one critical value for $x$" (factorisation, the formula or completing the square all accepted; "do not be concerned about the inequality symbol for the first two marks"); A1 both correct critical values; M1 "Chooses the outside region for their 2 critical values" (condones poor notation such as "$-\frac14\geq x\geq2$" at this mark only); A1 "Correct solution only" — $x\leq-\frac14$ or $x\geq2$, "oe" (or equivalent: set notation, "and", or two separate inequalities are all allowed forms).

Two real, quotable, genuinely new traps this pass recovers, not previously in this course's lesson or facts bank:

• The leading-coefficient factorisation trap, an explicit M0 special case in the real scheme: "Note that $4x^2-7x-2...0 \Rightarrow \left(x+\frac14\right)(x-2)...0 \Rightarrow x=...$ scores M0." Dropping the leading 4 into a unit-coefficient factorisation — $(x+\frac14)(x-2)$ instead of $(4x+1)(x-2)$ — produces the same two critical values by coincidence of this particular quadratic, but is not a valid factorisation of $4x^2-7x-2$ (expanding $(x+\frac14)(x-2)=x^2-1.75x-0.5$, not $4x^2-7x-2$) and the real scheme explicitly refuses the method mark for it, regardless of what the candidate writes next.
• The nonsensical-chain-notation trap, confirmed in both the mark scheme's own "do not allow" list and the examiner report's own words: the scheme explicitly disallows the final accuracy mark for $-\frac14\geq x\geq2$, and the examiner report states directly that "poor notation such as $-1/4 > x > 2$ lost many the final mark." This is a different failure from this topic's already-documented "between and outside reversed" trap: a candidate can correctly identify that the outside region is wanted and still lose the final mark by trying to force two disjoint branches into one continuous chain inequality, which asserts something that cannot be true of any real number at all ($x$ simultaneously less than $-\frac14$ and greater than $2$) — a notation failure independent of the underlying reasoning.

A third real, quoted finding, general rather than specific to this inequality: "a significant number either stated the roots or the inequalities, with no working", and separately "wrote down the incorrect factorisation $(x+\frac14)(x-2)$" — confirming the M0 special case above is not a theoretical mark-scheme edge case but a documented, real script error, on a paper that opens with an explicit no-calculator warning for exactly this reason.

#3.3 A region bounded by a line and a quadratic curve — a compound system in $x$ and $y$ — real anchors: Jan 2019 Q4 and Oct 2021 Q3(ii) [real]

The richest and most consistently-structured real sub-case for this topic — two independently confirmed sightings, both worth exactly 3 marks, both following the identical B1 (sight, no method) → M1 (solve for the missing numeric boundary) → A1 (complete, correctly-strict system) shape (facts bank §4.1, re-verified 2026-09-14 directly against the primary PDFs). The lesson file already teaches this skill in full, on VERIDIAN-original numbers; what follows here is both real anchors worked on their own real numbers, independently recomputed this pass — including one relationship (§ below) neither the lesson nor the facts bank derives.

Jan 2019, Q4 [real]. Line $l_1$: $2y=x$; curve $C$: $y=2x-\frac18x^2$; region $R$ bounded by $l_1$, $C$ and $l_2$ (parallel to the $y$-axis, through $C$'s positive $x$-intercept). $C$'s positive $x$-intercept: $2x-\frac18x^2=0 \Rightarrow x\left(2-\frac{x}{8}\right)=0 \Rightarrow x=16$. Real answer: $2y\leq x,\ y\geq2x-\frac18x^2,\ x<16$ — B1 (either inequality, sighted, "NB inequalities cannot be in terms of $R$"), M1 (solve $C$'s $x$-intercept), A1 (the complete system). Independently recomputed this pass, not stated in the real mark scheme itself: $l_1$ and $C$ actually cross each other at $\frac{x}{2}=2x-\frac{x^2}{8} \Rightarrow 4x=16x-x^2 \Rightarrow x^2-12x=0 \Rightarrow x=12$ — the region's true left-hand boundary, which the real scheme's own generosity clause ("allow $A\leq x<16$ where $A\leq12$") never requires a candidate to find, since 1.7's own guidance frames this question type around one calculated boundary, not two.

Oct 2021, Q3(ii) [real]. Curve $C$: $y=2x^2-50$; line $l$, gradient 3, meeting $C$ where $C$ crosses the negative $x$-axis. $C$'s negative $x$-intercept: $2x^2-50=0\Rightarrow x=\pm5$, so $(-5,0)$; line through it, gradient 3: $y-0=3(x+5)\Rightarrow y=3x+15$ — matching the real scheme's own quoted line exactly. Real answer: $y<2x^2-50,\ y>3x+15,\ x<a$ where $-5\leq a\leq6.5$ (M1 explicitly refuses a region-label answer such as "$R<2x^2-50$" here — "this scores M0", a method-mark failure, not merely a lost accuracy mark). Independently recomputed this pass, explaining a real scheme detail neither the facts bank nor the lesson derives: the generosity band $-5\leq a\leq6.5$ is not an arbitrary tolerance — it spans exactly between the two real $x$-values where line and curve intersect. Solving $2x^2-50=3x+15\Rightarrow2x^2-3x-65=0$: discriminant $=9-4(2)(-65)=9+520=529=23^2$, so $x=\frac{3\pm23}{4}=6.5$ or $-5$. The two boundary values a real candidate is credited for stating anywhere within are, independently verified, exactly the two genuine points where the two curves actually meet.

Both real examiner reports name the same failure mode independently: Jan 2019's "stating y < 16 was a frequent error" and "others stated inequalities involving R, which gained no credit"; Oct 2021's "little idea of how to approach the question" was common, and "far fewer achieved the final mark for the inequality in x" than picked up the first two marks — the third, calculated boundary is where both real series lose the most marks, not the two boundaries read straight off the sketch.

#3.4 A reciprocal inequality, reduced to a quadratic by multiplying through by $x^2$ — real anchor: Oct 2021 Q3(i) [real], spec's own named method

Spec 1.7's own guidance column (§1 above) names this reduction directly: $\frac{a}{x}<b$ becomes $ax<bx^2$. The real anchor uses it: "Solve $\frac3x>4$" [real, Oct 2021 Q3(i), 3 marks]. Worked on the real numbers: multiplying both sides by $x^2$ (always $\geq0$, so the inequality direction never flips, sidestepping any need to case-split on the sign of $x$) gives $3x>4x^2 \Rightarrow 4x^2-3x<0 \Rightarrow x(4x-3)<0$, critical values $x=0,\ \frac34$; $a=4>0$, so $<0$ is the between-the-roots case: $0<x<\frac34$.

The real mark scheme's most-credited route is this algebraic one, not a sketch — confirmed verbatim: "the few students who used the method of multiplying by x² usually achieved the correct inequality." The dominant real error was different: rearranging directly to $x<\frac34$ by implicitly treating $x$ as positive (dividing through by $x$ without splitting on its sign), earning only the mark scheme's own documented special case, "B1: States x < 3/4 only" — one mark out of three, for finding one of the two genuine critical values while silently losing the necessary $x>0$ condition. The examiner report's own diagnosis: "it was rare to see a sketch graph, which would have helped students to see that x > 0 was required."

#3.5 Representing an inequality graphically — shading and the dotted/solid convention — spec 1.8's own worked forms

Spec 1.8's guidance names two worked forms directly, "$y>x+r$" and "$y>ax^2+bx+c$" (§1 above), and states the dotted/solid requirement as non-optional. Honest note on the evidence: this course's WMA11 research has not found a specific, quoted examiner-report script error tied to the dotted/solid convention on its own, on this topic — it is taught here (and in the live lesson) as real, required spec content on the strength of 1.8's own verbatim wording, not left out for want of a matching quote, and not overstated as a documented trap it currently isn't.

[VERIDIAN, illustrative] Represent $y\geq-x^2+2x+3$. Roots of $-x^2+2x+3=0$: $-(x-3)(x+1)=0\Rightarrow x=3,-1$ (check: $-(x-3)(x+1)=-(x^2-2x-3)=-x^2+2x+3$ ✓). Boundary: solid curve (the inequality is non-strict, $\geq$), region above the curve shaded (a point's $y$-coordinate must be at least the curve's height there).

#3.6 Two already-solved inequalities combined on a number line — an honest gap

Spec 1.9's own wording is plural, "solutions of linear and quadratic inequalities" [together], and the live lesson's worked-chain and chain-drill blocks rehearse combining a solved quadratic inequality with a solved linear one via a number line, including the case where a whole branch of one region vanishes entirely against the other. No real WMA11 question combining two independently-solved inequalities in this specific way has been found by this course's research — the lesson's own closing flag states this plainly, and this document does not repeat that search with a different result. The nearest real evidence is Oct 2021 Q3(i)'s own diagnosis (§3.4 above), which is about a single reciprocal inequality losing a necessary case, not two combined regions; the lesson reuses that diagnosis honestly, for its general lesson only ("skipping the sketch is where a condition goes missing"), not as direct evidence that this exact combined-system shape has itself been examined.

#4. Synthesis: every real exam angle this specific topic has actually been tested from

Two structurally distinct trap families recur across all four real anchors: the final written line, not the algebra, is where marks are lost — a wrong variable or region label (Jan 2019), a dropped sign case from skipping a sketch (Oct 2021 Q3(i)), a region-label answer refused outright at the method-mark stage (Oct 2021 Q3(ii)), and a self-contradictory chain notation for a correctly identified region (June 2023) are four different surface errors converging on the same underlying lesson: correct critical-value algebra is necessary but not sufficient, and every real anchor here loses its last mark or two downstream of that algebra, not within it.

#5. Honest limits of the evidence

WMA11-verified-facts.md §6 does not list this topic among those with "enough verified material for a genuine standalone lesson" — it sits instead among the fuller §4.1 material, on genuinely thinner evidence than the six lessons built from that shortlist. signal-frequency-and- boundaries.md's own frequency table reports the compound-region skill (§3.3 above) as confirmed in 2 of 5 fully-mined WMA11 examiner-report series (Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025) — a genuine, repeatable count, not an estimate. The reciprocal-reduction anchor (§3.4) sits on the same Oct 2021 paper but is, strictly, spec 1.11 reciprocal-function territory reached via 1.7's own authorised method, not counted a second time in that "2 of 5" figure — this document keeps that distinction exactly as the signal brief itself draws it, rather than inflating the apparent count by merging them. The rearrangement anchor (§3.2, June 2023 Q1) sits outside the five fully-mined series entirely — it was a targeted, single-question spot-check in the facts bank's own 2026-09-25 pass, and this document is the first place in this course to carry its full mark-scheme and examiner-report detail rather than the one-line summary that pass recorded. Two series (Oct 2022, Jan 2024) are confirmed genuine but were never mined question-by-question, so this topic's presence or absence in them is simply unknown. §3.5 (representing graphically) and §3.6 (combined inequalities) are stated above as real spec content with no matching real trap or anchor found — not as documented weak points, and not omitted for being thin. §2's two boundary cases are marked [VERIDIAN, illustrative] because no real anchor applying $\Delta=0$ or $\Delta<0$ to a quadratic inequality question (as opposed to the parameter-constraining use of the discriminant covered by a different lesson) has been found. No likelihood figure or grade boundary is stated or implied anywhere in this document.

#6. What this document deliberately does not repeat

The lesson file's three prequestions, both teach blocks' full prose, its mechanism block's own worked derivation, its two-curve diagram, its complete worked-chain and chain-drill (both VERIDIAN-original combined-inequality systems), its two full marked-solutions and their commonWrongPath variants, its seven-item trap-taxonomy, its self-explain prompt, its beyond-spec cubic extension, its six MCQs, reference card and closing flag all already exist in quadratic-inequalities-graphically.ts and are not reconstructed here. codex-formula-and-method- bank.md §1.1.5–§1.1.6 already states the general three-route method-mark convention and the reciprocal-inequality identity as compact reference; this document does not restate those, only adds the specific June 2023 M0 caveat on top of the general three-route rule, and the guidance- column text neither of those sources had yet transcribed. This document's own purpose was: the spec's guidance-column wording verified fresh against the primary PDF (§1); the mechanism as compact algebra rather than taught prose (§2); a complete map of every real, distinct sub-case this evidence base supports, several worked on numbers no other document in this course has used (§3); and the citation synthesis that sits alongside — not beneath — all of the above (§4–§5).