ird inequality","content":"Now we determine the inequality for the line $y = 2x + 2$:\n\n- The shaded region lies **below** this solid boundary line.\n- Therefore, the $y$-values must be less than or equal to $2x + 2$:\n\n$$y \\leq 2x + 2$$"},{"type":"text","order":5,"title":"State final answer","content":"Combining all three boundary inequalities, we define the shaded region $R$ by:\n\n$$\\begin{aligned} x &\\leq 4 \\\\ y &\\geq -x \\\\ y &\\leq 2x + 2 \\end{aligned}$$"}]}],"generatedAt":"2026-06-13T04:48:22.076Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"d077d9c2-c39b-40ca-be91-ac362bf394df","revisionId":"ad54aefe-c159-4c46-a0a5-0363547d1eea","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"bc7084bd-46ad-4606-9440-67e788924d89","label":"a","content":"On the grid shade the region that satisfies all these inequalities\n\n$$y < 3 - x$$\n\n$$y > 2x - 3$$\n\n$$y > -1$$\n\nLabel the region **R**.","markscheme":"- Draws lines $y = 3 - x$, $y = 2x - 3$, and $y = -1$ (all dashed). M1\n- Correct shading of the region bounded by the three lines. M1\n- Region correctly labeled **R**. A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Accurately draws the boundary lines $y = 3 - x$, $y = 2x - 3$, and $y = -1$ as dashed lines.","code":"M"},{"id":"p2","marks":1,"description":"Correctly identifies and shades the region satisfying all three inequalities.","code":"M"},{"id":"p3","marks":1,"description":"Label the shaded region with the letter 'R'.","code":"A"}],"finalAnswers":["Region bounded by (1, -1), (4, -1), and (2, 1)"],"generatedAt":"2026-05-06T14:01:08.449Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"graphical_inequalities","label":"Draw, test and shade","steps":[{"type":"text","order":0,"title":"Replace each inequality","content":"Replace each strict inequality sign with an equals sign to get its boundary:\n\n$$y=3-x,\\qquad y=2x-3,\\qquad y=-1.$$\n\nBecause all the inequalities are strict ($<$ or $>$), none of the boundary lines belongs to the solution, so draw every boundary as a **dashed** line.","highlights":[{"text":"$$$y < 3 - x$$","location":"specification"},{"text":"$$$y > 2x - 3$$","location":"specification"},{"text":"$$$y > -1$$","location":"specification"}]},{"type":"graph","order":1,"title":"Draw the boundaries","viewport":{"top":7,"left":-3,"right":6,"bottom":-3},"axisLabels":{"x":"x","y":"y"},"description":"Draw all three boundaries as dashed lines because every inequality is strict: red is $y=3-x$, blue is $y=2x-3$, and green is $y=-1$. Drawing all three correctly as dashed lines earns M1.","expressions":[{"id":"expr-0","color":"red","latex":"y=3-x","lineStyle":"dashed"},{"id":"expr-1","color":"sky","latex":"y=2x-3","lineStyle":"dashed"},{"id":"expr-2","color":"green","latex":"y=-1","lineStyle":"dashed"}]},{"type":"text","order":2,"title":"Test a point","content":"Use the test point $(0,0)$. It checks which side of each boundary is wanted:\n\n$$0<3-0=3$$\n$$0>2(0)-3=-3$$\n$$0>-1$$\n\nAll three statements are true, so the required side is the side containing $(0,0)$ for each line. The overlap extends indefinitely to the left. It lies above $y=-1$ until that line meets $y=2x-3$ at $(1,-1)$, then above $y=2x-3$ until it meets $y=3-x$ at $(2,1)$, and remains below $y=3-x$ throughout."},{"type":"graph","order":3,"title":"Shade the overlap","viewport":{"top":7,"left":-3,"right":6,"bottom":-3},"axisLabels":{"x":"x","y":"y"},"description":"Shade only the common overlap. It is above $y=-1$ and below $y=3-x$ up to $x=1$; from $x=1$ to $x=2$, it is above $y=2x-3$ and below $y=3-x$. The inequalities are strict, so the three boundary lines stay dashed. Correct shading earns the second M1.","expressions":[{"id":"expr-0","color":"amber","latex":"-1

Inequalities on Graphs

Question 3

Inequalities on Graphs Questions

Inequalities on Graphs

Question 3

Inequalities on Graphs Questions