":"v3"},"options":[]},{"id":"1f77f7dd-3612-4517-bdc9-1427b6e22828","label":"c","content":"Solve $6x + 8 \\leq 2x + 28$","markscheme":"- Subtracts $2x$ from both sides: $4x + 8 \\leq 28$ M1\n- Subtracts 8 from both sides: $4x \\leq 20$ M1\n- Final answer: $x \\leq 5$ A1","marks":3,"order":2,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Subtracts $2x$ from both sides resulting in $4x + 8 \\leq 28$.","code":"M"},{"id":"p2","marks":1,"description":"Subtracts 8 from both sides resulting in $4x \\leq 20$.","code":"M"},{"id":"p3","marks":1,"description":"Correct final inequality: $x \\leq 5$.","code":"A"}],"finalAnswers":["x \\leq 5"],"generatedAt":"2026-05-06T14:03:26.696Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"balance_both_sides","label":"Balance both sides","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=original}{\\color{@sky}{6x+8\\leq 2x+28}}$"},{"latex":"$$\\htmlData{anchor=subtract_both}{\\color{@sky}{6x+8-2x\\leq 2x+28-2x}}$"},{"latex":"$$\\htmlData{anchor=first_result}{\\color{@sky}{4x+8\\leq 28}}$"}],"connections":[{"to":"subtract_both","from":"original","color":"sky","style":"solid"},{"to":"first_result","from":"subtract_both","color":"sky","label":"simplify","style":"solid"}]},"order":1,"title":"Subtract the x-terms","description":"Subtract 2x from both sides. This keeps the inequality balanced and earns the first method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=before_subtract}{4x+8\\leq 28}$"},{"latex":"$$\\htmlData{anchor=subtract_eight}{4x+8-8\\leq 28-8}$"},{"latex":"$$\\htmlData{anchor=after_subtract}{4x\\leq 20}$"}],"connections":[{"to":"subtract_eight","from":"before_subtract","color":"orange"},{"to":"after_subtract","from":"subtract_eight","color":"orange","label":"simplify"}]},"order":2,"title":"Subtract the constant","description":"Now subtract 8 from both sides. Reaching 4x ≤ 20 earns the second method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=before_divide}{\\color{@green}{4x\\leq 20}}$"},{"latex":"$$\\htmlData{anchor=divide_four}{\\color{@green}{\\dfrac{4x}{4}\\leq \\dfrac{20}{4}}}$"},{"latex":"$$\\htmlData{anchor=final_answer}{\\color{@green}{x\\leq 5}}$"}],"connections":[{"to":"divide_four","from":"before_divide","color":"green","label":"divide both sides","style":"solid"},{"to":"final_answer","from":"divide_four","color":"green","label":"simplify","style":"solid"}]},"order":3,"title":"Divide by four","description":"Divide both sides by positive 4. Because the divisor is positive, the inequality sign stays in the same direction. The correct final answer earns A1."}]}],"generatedAt":"2026-07-19T16:53:27.812Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"b8e6b030-0eb3-4d74-982b-d2276ea9423d:direct_graphical","label":"(a) Direct Graphical Representation","steps":[{"type":"text","order":0,"title":"Analyze the inequality","content":"We need to represent the inequality $n < 5$ on the given number line.\n\nLet's break down what $n < 5$ means:\n* The symbol $<$ means **\"less than\"**.\n* This tells us that $n$ can be any number smaller than $5$, but it **cannot** be equal to $5$ itself.","highlights":[{"text":"$$n < 5$","location":"specification"}]},{"type":"text","order":1,"title":"Mark the boundary value","content":"To represent that $5$ is **not included** in our values for $n$, we place an **open (unfilled) circle** directly at $5$ on the number line.\n\n$$\\text{Open circle at } 5 \\implies 5 \\text{ is excluded}$$"},{"type":"text","order":2,"title":"Draw the direction line","content":"Since $n$ is **less than** $5$, we need to point towards all the numbers that are smaller than $5$ (to the left).\n\nDraw a straight line starting from the open circle at $5$ extending to the left, ending with an arrow to show it continues indefinitely in that direction."},{"type":"text","order":3,"title":"Verify markscheme requ

Inequalities Questions

Inequalities Questions

Inequalities Questions

Inequalities Questions