,"title":"Select the non-positive region","viewport":{"top":8,"left":-5,"right":4,"bottom":-14},"axisLabels":{"x":"x","y":"y"},"description":"The coefficient of x^2 is positive, so the parabola opens upwards. It is below or on the x-axis between its roots; the solid root points are included because the inequality is <= 0. Choosing this region and both endpoints earns the final mark. Therefore, -4 <= x <= 3.","expressions":[{"id":"expr-0","color":"sky","latex":"y=x^2+x-12"},{"id":"expr-1","color":"green","latex":"x^2+x-12\\leq y\\leq0\\left\\{-4\\leq x\\leq3\\right\\}"},{"id":"expr-2","color":"teal","latex":"x=-4\\left\\{-12.25\\leq y\\leq0\\right\\}","lineStyle":"dashed"},{"id":"expr-3","color":"teal","latex":"x=3\\left\\{-12.25\\leq y\\leq0\\right\\}","lineStyle":"dashed"},{"id":"expr-4","color":"green","label":"x=-4","latex":"(-4,0)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-5","color":"green","label":"x=3","latex":"(3,0)","pointStyle":"point","labelOrientation":"left"},{"id":"expr-6","color":"green","label":"-4 <= x <= 3","latex":"(-0.5,-5)","pointStyle":"point","labelOrientation":"above"}]}]}],"generatedAt":"2026-07-21T04:09:55.250Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":84}},{"id":"f22a489f-86ee-4e89-b706-d4d5bf360a18","revisionId":"5ffaef70-7936-44a8-b17f-8188db4d59ec","questionNumber":"10","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Question Set 3: Equations and Inequalities","calculatorUse":null,"parts":[{"id":"fd8f6ee4-bd64-4d40-ba78-11a1b5b10519","label":"0","content":"Solve the inequality $x^2 - 3x > 18$","markscheme":"$$x^2 - 3x - 18 > 0$\n$(x - 6)(x + 3) > 0$\nCritical values: $x = 6, x = -3$\n![](https://assets.mathsgenie.co.uk/booklet/1bb99f9b0d2384a94e00e93132ceb2ba77a846d9acf81b31fc1442a69a1efe1d.jpg)\n$x < -3$ or $x > 6$ 2","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds critical values $-3$ and $6$.","code":"M"},{"id":"p2","marks":1,"description":"Correct final inequality $x < -3$ or $x > 6$.","code":"A"}],"finalAnswers":["x < -3 \\text{ or } x > 6"],"generatedAt":"2026-05-12T02:14:06.552Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"factor_sign_chart","label":"Factorise and use signs","steps":[{"type":"text","order":0,"title":"Rearrange the inequality","content":"Move $18$ to the left by subtracting $18$ from both sides:\n\n$$x^2-3x-18>0$$\n\nNow the inequality is compared with zero, ready to factorise.","highlights":[{"text":"Solve the inequality $x^2 - 3x > 18$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=expanded}{\\color{@sky}{x^2-3x-18>0}}"},{"latex":"\\htmlData{anchor=factorised}{\\color{@green}{(x-6)(x+3)>0}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"factorised"}],"connections":[{"to":"factorised","from":"expanded","color":"green","label":"factorise","style":"solid"}]},"order":1,"title":"Factorise the quadratic","description":"Factorise the quadratic. Find two numbers with product -18 and sum -3: -6 and 3. Write the factorised inequality clearly before analysing its sign. Examiner annotation: this is the credited factorisation method."},{"type":"text","order":2,"title":"Find critical values","content":"Set each factor equal to zero:\n\n$$x-6=0\\quad\\text{or}\\quad x+3=0$$\n\nSo the critical values are $x=6$ and $x=-3$. These values split the number line into three intervals."},{"type":"text","order":3,"title":"Test the three intervals","content":"Test one value in each interval in $(x-6)(x+3)$:\n\n$$\\begin{array}{c|c}\n\\text{interval}&\\text{sign of }(x-6)(x+3)\\\\ \\hline\nx<-3&\\text{positive}\\\\\n-36&\\text{positive}\n\\end{array}$$\n\nFor example, the expression is positive at $x=-4$ and at $x=7$, but negative at $x=0$. We need the positive intervals because the inequality is $>0$."},{"

Equations and Inequalities

Question 1

Equations and Inequalities Questions

Equations and Inequalities

Question 1

Equations and Inequalities Questions