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Because the inequalities are strict, points on either line or on the $y$-axis are not allowed.","highlights":[{"text":"$$y > x + 3$","location":"specification"},{"text":"$$y + x < 8$","location":"specification"},{"text":"$$x > 0$","location":"specification"}]},{"type":"graph","order":1,"title":"Find the valid region","viewport":{"top":8.5,"left":-0.2,"right":8.5,"bottom":-0.2},"axisLabels":{"x":"x","y":"y"},"description":"The valid region is above $y=x+3$, below $y+x=8$, and to the right of the $y$-axis. The boundaries are dashed because the inequalities are strict, so points on a boundary do not count.","expressions":[{"id":"expr-0","color":"green","latex":"x>0\\left\\{y>x+3\\right\\}\\left\\{y<8-x\\right\\}"},{"id":"expr-1","color":"sky","latex":"y=x+3\\left\\{0\\le x\\le5\\right\\}","lineStyle":"dashed"},{"id":"expr-2","color":"orange","latex":"y=8-x\\left\\{0\\le x\\le8\\right\\}","lineStyle":"dashed"},{"id":"expr-3","color":"sky","label":"y=x+3","latex":"\\left(4.5,7.5\\right)","hidden":true,"labelOrientation":"above"},{"id":"expr-4","color":"orange","label":"y+x=8","latex":"\\left(6.5,1.5\\right)","hidden":true,"labelOrientation":"above"}]},{"type":"graph","order":2,"title":"Mark integer points","viewport":{"top":8.5,"left":-0.2,"right":8.5,"bottom":-0.2},"axisLabels":{"x":"x","y":"y"},"description":"Scan integer-coordinate grid points only. At $x=1$, the strict inequalities give $40\\right\\}"},{"id":"expr-1","color":"sky","latex":"y=x+3\\left\\{0\\le x\\le5\\right\\}","lineStyle":"dashed"},{"id":"expr-2","color":"orange","latex":"y=8-x\\left\\{0\\le x\\le8\\right\\}","lineStyle":"dashed"},{"id":"expr-3","color":"green","latex":"x=0\\left\\{3\\le y\\le8\\right\\}","lineStyle":"dashed"},{"id":"expr-4","color":"red","label":"(1, 5)","latex":"(1,5)","pointStyle":"cross","labelOrientation":"right"},{"id":"expr-5","color":"red","label":"(1, 6)","latex":"(1,6)","pointStyle":"cross","labelOrientation":"above"}]}]}],"generatedAt":"2026-07-20T11:35:47.948Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"22d463a4-cedb-44dc-b60c-d4e3f0a6fa75","revisionId":"bbf373ec-6c07-4ce0-92a7-9384a2ad8e37","questionNumber":"4","questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":3,"sourceTitle":"Inequalities on Graphs","calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"a4d2b30b-7b5d-4251-817b-efc92afb4e4d","label":"","content":"On the grid shade the region that satisfies all these inequalities\n\n$$x < 3$$\n\n$$y > -2$$\n\n$$y \\leq 2x - 1$$\n\nLabel the region **R**.\n\n![Coordinate grid](https://assets.mathsgenie.co.uk/booklet/9a7eb1f09a4d9e343521a37851556ed3f48dc4d166e63ee40ce35a1132c920d5.jpg)","markscheme":"[Corrected solution grid: shade the region satisfying $-\\frac{1}{2}M1\n- Correct shading of the region bounded by the lines. M1\n- Region correctly labeled **R**. A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Draws the three boundary lines correctly, using dashed lines for strict inequalities ($x < 3$, $y > -2$).","code":"M"},{"id":"p2","marks":1,"description":"Shades the correct region bounded by the three lines.","code":"M"},{"id":"p3","marks":1,"description":"Labels the correct region with 'R'.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-05T15:37:52.705Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":{"methods":[{"id":"a4d2b30b-7b5d-4251-817b-efc92afb4e4d:boundary_analysis","label":"Direct Boundary Analysis","steps":[{"type":"text","order":0,"title":"Understand the boundary rules","content":"To represent inequalities on a graph, we start by looking at each inequality symbol to decide if the boundary line is **solid** or **dashed**:\n\n* Use a **dashed line** for strict inequalities ($<$ or $>$), because the points on the boundary line are not included in the region.\n* Use a **solid line** for non-strict inequalities ($\\le$ or $\\ge$), because the points on the boundary line are included."},{"type":"text","order":1,"title":"Plot the line $x < 3$","content":"First, we look at the inequality $x < 3$:\n* The boundary line is $x = 3$, which is a vertical line passing through $3$ on the $x$-axis.\n* Because the inequality is strict ($<$), we draw this as a **dashed line**.\n* Since we want values where $x$ is less than $3$, the correct region lies to the **left** of this line.","highlights":[{"text":"x < 3","location":"specification"}]},{"type":"text","order":2,"title":"Plot the line $y > -2$","content":"Next, we look at the inequality $y > -2$:\n* The boundary line is $y = -2$, which is a horizontal line passing through $-2$ on the $y$-axis.\n* Because the inequality is strict ($>$), we draw this as a **dashed line**.\n* Since we want values where $y$ is greater than $-2$, the correct region lies **above** this line.","highlights":[{"text":"y > -2","location":"specification"}]},{"type":"text","order":3,"title":"Plot the line $y \\leq 2x - 1$","content":"Now, we look at the linear inequality $y \\leq 2x - 1$:\n* The boundary line is $y = 2x - 1$, which has a $y$-intercept of $-1$ and a gradient of $2$.\n* Plot a few points to draw this line:\n * When $x = 0$, $y = -1 \\implies (0, -1)$\n * When $x = 1$, $y = 1 \\implies (1, 1)$\n * When $x = 2$, $y = 3 \\implies (2, 3)$\n * When $x = 3$, $y = 5 \\implies (3, 5)$\n* Because the inequality is non-strict ($\\le$), we draw a **solid line** through these points.\n* Since we want values where $y$ is less than or equal to $2x - 1$, the correct region lies **below** (or to the right of) this line.","highlights":[{"text":"y \\leq 2x - 1","location":"specification"}]},{"type":"text","order":4,"title":"Shade and label region R","content":"By combining all three conditions, we find the enclosed triangular region that is:\n1. To the **left** of the dashed line $x = 3$\n2. **Above** the dashed line $y = -2$\n3. **Below** the solid line $y = 2x - 1$\n\nShade this region on your grid and label it clearly with the letter **R** to secure all 3 marks.\n\n$$\\begin{aligned}\n\\text{Vertices of region } R: & \\quad (-0.5, -2) \\quad [\\text{intersection of } y=-2 \\text{ and } y=2x-1] \\\\\n& \\quad (3, -2) \\quad [\\text{intersection of } x=3 \\text{ and } y=-2] \\\\\n& \\quad (3, 5) \\quad [\\text{intersection of } x=3 \\text{ and } y=2x-1]\n\\end{aligned}$$"}]},{"id":"a4d2b30b-7b5d-4251-817b-efc92afb4e4d:test_point","label":"Test Point Verification","steps":[{"type":"text","order":0,"title":"Identify boundary lines","content":"To solve this graphical inequality problem, we first need to plot the boundary lines corresponding to each inequality:\n\n1. For $$x < 3$$, the boundary line is $$x = 3$$. Since the inequality is strict ($<$), we draw this as a **dashed line**.\n2. For $$y > -2$$, the boundary line is $$y = -2$$. Since this is also strict ($>$), we draw this as a **dashed line**.\n3. For $$y \\leq 2x - 1$$, the boundary line is $$y = 2x - 1$$. Since the inequality includes 'equal to' ($\\leq$), we draw this as a **solid line**.","highlights":[{"text":"x < 3","location":"specification"},{"text":"y > -2","location":"specification"},{"text":"y \\leq 2x - 1","location":"specification"}]},{"type":"text","order":1,"title":"Plot the line y = 2x - 1","content":"To plot the line $$y = 2x - 1$$, we can find a few points by substituting convenient $x$-values:\n\n* If $$x = 0$$, then $$y = 2(0) - 1 = -1$$ $\\rightarrow$ Point $$(0, -1)$$\n* If $$x = 2$$, then $$y = 2(2) - 1 = 3$$ $\\rightarrow$ Point $$(2, 3)$$\n* If $$x = 3$$, then $$y = 2(3) - 1 = 5$$ $\\rightarrow$ Point $$(3, 5)$$\n\nDraw a straight **solid line** passing through these points."},{"type":"text","order":2,"title":"Test point (0,0) for x < 3","content":"Now we will use the **Test Point Verification** method to find which side of each boundary line to shade. A convenient test point is $$(0,0)$$, as it does not lie on any of the boundary lines.\n\nLet's test $$(0,0)$$, where $$x = 0$$ and $$y = 0$$, in the first inequality:\n\n$$0 < 3 \\quad \\text{(True)}$$\n\nSince this is **true**, the region satisfying $$x < 3$$ is the side of the vertical line $$x = 3$$ that **contains** the origin $$(0,0)$$ (the left side)."},{"type":"text","order":3,"title":"Test point (0,0) for y > -2","content":"Now let's test the point $$(0,0)$$ in the second inequality:\n\n$$0 > -2 \\quad \\text{(True)}$$\n\nSince this is also **true**, the region satisfying $$y > -2$$ is the side of the horizontal line $$y = -2$$ that **contains** the origin $$(0,0)$$ (the region above the line)."},{"type":"text","order":4,"title":"Test point (0,0) for y <= 2x - 1","content":"Now let's test the point $$(0,0)$$ in the third inequality:\n\n$$0 \\leq 2(0) - 1$$\n$$0 \\leq -1 \\quad \\text{(False)}$$\n\nSince this is **false**, the region satisfying $$y \\leq 2x - 1$$ is the side of the line $$y = 2x - 1$$ that **does not contain** the origin $$(0,0)$$ (the bottom-right side of the line)."},{"type":"text","order":5,"title":"Shade and label the region R","content":"We now combine these results to identify the overlapping region:\n* To the **left** of the dashed line $$x = 3$$\n* **Above** the dashed line $$y = -2$$\n* **Below/right** of the solid line $$y = 2x - 1$$\n\nShade this triangular region where all three conditions are satisfied, and clearly label it with the letter **R** to get full marks.","highlights":[{"text":"Label the region R.","location":"specification"}]}]}],"generatedAt":"2026-06-12T11:47:48.333Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"gcse_maths__ccea","summaries":[{"id":"3d479758-a041-4008-bcf8-921ef9291ca8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"c1999271-0415-4ae5-83fe-87c951004af7","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"d077d9c2-c39b-40ca-be91-ac362bf394df","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"d3ec274c-b8ed-4604-acdf-8a0b50deafbb","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"e1eda067-4ab9-4c62-942b-864281ae27f6","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"f4a4460c-2b0d-4648-bd2d-89c82768f1e4","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"fa10453a-c1bf-4b3a-8f58-7dd32c8264a2","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"a382e96b-1fe0-4a66-a000-9cc66f0298be","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"018e0fef-258b-4130-ab9c-108a1acd8d26","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"04b6dab4-947e-44ff-95d0-ef50bd56cb0c","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"4ca79549-9e2d-4594-99bf-a346bc726f12","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"54e28262-f019-4a26-997d-4718312523ee","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"7f717622-0b5f-428e-a63d-99f9fe3ac5ed","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"c90edb37-db8d-4722-9d7c-0217042c4530","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"fc49b4d7-fb13-4d63-aa71-1ab0244f0a2c","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"6d24b22f-d065-426b-ac38-ce273717897d","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"075f5334-fa95-47bb-811d-09178a5493aa","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"af5b1546-a5e7-459c-a115-f3eef0d4b028","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"bd5ce91b-7f96-4a31-ae4f-c729c559279c","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"c4f0f499-5b3c-4feb-a9b7-e5098bef5763","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"e34cdc3c-d9fd-4e8d-8f2a-9847b681b972","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"eca836c8-d55d-4c98-8ef2-1f59d001778f","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"fc73d8e8-965a-4082-a4e7-5396f22f82db","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"22d463a4-cedb-44dc-b60c-d4e3f0a6fa75","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"e151042d-a057-41fb-8af3-ea8a826313f5","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"a9a2e79a-71ae-4eca-8e3a-bc1229b9793b","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"b601c3d3-d6ac-42c1-be2a-d30a65d32980","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":false}],"title":"Inequalities on Graphs","topicId":"09b191c9-ec62-c7bc-357a-f042010e53ad","topicSidebar":"$L3a","topicSlug":"grade-6-6-7-inequalities-on-graphs","topicTitle":"Inequalities on Graphs"}]}],"$L3b","$L3c"]}]

Inequalities on Graphs

Question 16

Inequalities on Graphs Questions

Inequalities on Graphs

Question 16

Inequalities on Graphs Questions