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grid](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/3d479758-a041-4008-bcf8-921ef9291ca8/image-1-f871f03b4a43a514c37d2e89.svg)","markscheme":"- Draws lines $x = -2$ (solid), $y = 3$ (dashed), and $y = x + 1$ (dashed). M1\n- Correct shading of the region bounded by the lines. M1\n- Region correctly labeled **R** with correct boundary line styles. 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The type of inequality sign determines whether the line is solid or dashed:\n\n* **Solid line** ($\\geq$ or $\\leq$): The boundary points are included in the region.\n* **Dashed line** ($<$ or $>$): The boundary points are not included.\n\nLet's identify our three boundary lines:\n1. For $x \\geq -2$, the boundary line is $x = -2$ (a **solid** vertical line).\n2. For $y < 3$, the boundary line is $y = 3$ (a **dashed** horizontal line).\n3. For $y > x + 1$, the boundary line is $y = x + 1$ (a **dashed** diagonal line).","highlights":[{"text":"x \\geq -2","location":"specification"},{"text":"y < 3","location":"specification"},{"text":"y > x + 1","location":"specification"}]},{"type":"text","order":1,"title":"Draw the boundary lines","content":"Let's place these lines on the coordinate grid:\n\n* **Line 1 ($x = -2$):** Draw a solid vertical line passing through $-2$ on the $x$-axis.\n* **Line 2 ($y = 3$):** Draw a dashed horizontal line passing through $3$ on the $y$-axis.\n* **Line 3 ($y = x + 1$):** To draw this line, plot a few coordinate points:\n * When $x = -2$, $y = -2 + 1 = -1 \\rightarrow (-2, -1)$\n * When $x = 0$, $y = 0 + 1 = 1 \\rightarrow (0, 1)$\n * When $x = 2$, $y = 2 + 1 = 3 \\rightarrow (2, 3)$\n \n Draw a dashed diagonal line passing through these points."},{"type":"text","order":2,"title":"Apply directional reasoning","content":"Now we look at the directions indicated by each inequality to find where they overlap:\n\n1. **$x \\geq -2$**: The $x$-values must be greater than or equal to $-2$. This means the region lies to the **right** of the vertical line $x = -2$.\n2. **$y < 3$**: The $y$-values must be less than $3$. This means the region lies **below** the horizontal line $y = 3$.\n3. **$y > x + 1$**: The $y$-values must be greater than $x + 1$. This means the region lies **above** the diagonal line $y = x + 1$."},{"type":"text","order":3,"title":"Shade and label region R","content":"Find the triangular region that satisfies all three directional conditions simultaneously:\n\n* To the **right** of $x = -2$\n* **Below** $y = 3$\n* **Above** $y = x + 1$\n\nThis is the triangular region bounded by the vertices $(-2, -1)$, $(-2, 3)$, and $(2, 3)$.\n\nShade this region on your grid and clearly label it with the letter **R** to complete the question.","highlights":[{"text":"Label the region R.","location":"specification"}]}]},{"id":"96023108-cae0-4f9f-8af7-c51795d718d8:test_points","label":"Test Point Verification","steps":[{"type":"text","order":0,"title":"Identify the boundary lines","content":"To graph the inequalities, first determine the equation and style of each boundary line:\n\n* **$x = -2$**: A **solid** vertical line (since the inequality $x \\geq -2$ includes the \"or equal to\" relation).\n* **$y = 3$**: A **dashed** horizontal line (since the inequality $y < 3$ is strict).\n* **$y = x + 1$**: A **dashed** diagonal line with a $y$-intercept of $(0, 1)$ and gradient of $1$ (since the inequality $y > x + 1$ is strict)."},{"type":"text","order":1,"title":"Plot the lines","content":"Draw these three boundary lines on the grid:\n\n1. Plot the vertical solid line at $x = -2$.\n2. Plot the horizontal dashed line at $y = 3$.\n3. Plot the dashed diagonal line passing through the points $(-1, 0)$, $(0, 1)$, and $(1, 2)$ to represent $y = x + 1$."},{"type":"text","order":2,"title":"Verify with a test point","content":"Choose a convenient test point inside the region enclosed by these lines, such as $(0, 2)$, to verify if it satisfies all three inequalities:\n\n* **For $x \\geq -2$**:\n $$0 \\geq -2 \\quad \\text{(True)}$$\n\n* **For $y < 3$**:\n $$2 < 3 \\quad \\text{(True)}$$\n\n* **For $y > x + 1$**:\n $$2 > 0 + 1 \\implies 2 > 1 \\quad \\text{(True)}$$\n\nSince the point $(0, 2)$ satisfies all three inequalities, this is the correct region to shade."},{"type":"text","order":3,"title":"Shade and label region R","content":"Shade the triangular region containing the test point $(0, 2)$ that is bounded by the three lines:\n* To the right of the solid line $x = -2$\n* Below the dashed line $y = 3$\n* Above the dashed line $y = x + 1$\n\nClearly write the letter **R** inside this shaded area to complete the graph."}]}],"generatedAt":"2026-06-12T14:23:01.545Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":32},"walkthroughVideo":null},{"id":"d077d9c2-c39b-40ca-be91-ac362bf394df","revisionId":"ad54aefe-c159-4c46-a0a5-0363547d1eea","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"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**.\n\n![Coordinate grid](https://assets.mathsgenie.co.uk/figures/coords/fa8177263de49229c5349d43115a65fe4cf67bf742d1838adfdf4a0c651b1614.png)","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\n\n![Solution](https://assets.mathsgenie.co.uk/figures/coords/fccfd61be37773a222f48da51c2cad4c32356599d7b601c219aa5607c6c931e7.png)","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"},"tableImages":[],"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