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grid](https://assets.mathsgenie.co.uk/figures/coords/2b8e621edc427f0def38040590cd9da914d6b4af1c5e1c45f6ce50fb7c3b318d.png)","markscheme":"- Draws lines $y = 4 - 2x$, $y = x + 1$, and $y = -1$ (all dashed). M1\n- Correct shading of the region satisfying all three inequalities. 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 $y = 4 - 2x$, $y = x + 1$, 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 (-2, -1), (2.5, -1), and (1, 2)"],"generatedAt":"2026-05-06T14:01:10.135Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":{"methods":[{"id":"fce482a9-8595-4df9-bd98-690b15c99780:direct_analysis","label":"Direct Boundary Analysis and Shading","steps":[{"type":"text","order":0,"title":"Identify the boundary lines","content":"To represent the given inequalities on a grid, we must first determine their boundary lines. Because all three inequalities contain strict inequality signs ($<$ or $>$), all boundary lines must be drawn as **dashed lines**:\n\n1. For $y < 4 - 2x$, the boundary line is $$y = 4 - 2x$$\n2. For $y > x + 1$, the boundary line is $$y = x + 1$$\n3. For $y > -1$, the boundary line is $$y = -1$$"},{"type":"text","order":1,"title":"Plot the boundary lines","content":"Let's find key coordinates to plot each dashed line accurately:\n\n* **Line 1 ($y = 4 - 2x$):** \n * If $x = 0$, $y = 4$ giving the point $(0, 4)$.\n * If $x = 2$, $y = 0$ giving the point $(2, 0)$.\n * Plot a dashed straight line passing through these points.\n\n* **Line 2 ($y = x + 1$):**\n * If $x = 0$, $y = 1$ giving the point $(0, 1)$.\n * If $x = 2$, $y = 3$ giving the point $(2, 3)$.\n * Plot a dashed straight line passing through these points.\n\n* **Line 3 ($y = -1$):**\n * This is a horizontal dashed line passing through all points where the $y$-coordinate is $-1$ (e.g., $(0, -1)$)."},{"type":"text","order":2,"title":"Analyze the inequality regions","content":"Using the **Direct Boundary Analysis** approach, we interpret the inequality signs directly to locate our target region:\n\n* **$y < 4 - 2x$**: Since $y$ is 'less than' the expression, our region lies **below** the line $y = 4 - 2x$.\n* **$y > x + 1$**: Since $y$ is 'greater than' the expression, our region lies **above** the line $y = x + 1$.\n* **$y > -1$**: Since $y$ is 'greater than' $-1$, our region lies **above** the horizontal line $y = -1$."},{"type":"text","order":3,"title":"Shade and label region R","content":"Now, locate the region on your grid where all three conditions are satisfied simultaneously:\n\n* It is **below** the steep line going down to the right ($y = 4 - 2x$).\n* It is **above** the line going up to the right ($y = x + 1$).\n* It is **above** the horizontal line ($y = -1$).\n\nThe overlapping region is **unbounded to the left**, rather than triangular. Shade the entire region satisfying all three inequalities on your grid and clearly label it with the letter **R** to secure all 3 marks."}]},{"id":"fce482a9-8595-4df9-bd98-690b15c99780:test_points","label":"Using Test Points","steps":[{"type":"text","order":0,"title":"Identify boundary lines","content":"To represent the inequalities on a grid, we must first identify the corresponding boundary lines. Because all three inequalities contain strict inequality symbols ($<$ and $>$, rather than $\\le$ or $\\ge$), all our boundary lines must be drawn as **dashed lines**:\n\n1. $$y = 4 - 2x$$\n2. $$y = x + 1$$\n3. $$y = -1$$","highlights":[{"text":"$$y < 4 - 2x$","location":"specification"},{"text":"$$y > x + 1$","location":"specification"},{"text":"$$y > -1$","location":"specification"}]},{"type":"text","order":1,"title":"Plot the boundary lines","content":"Let's find coordinates to help draw each dashed line on the grid:\n\n* **For $y = 4 - 2x$:**\n * If $x = 0$, $y = 4$ $\\rightarrow (0, 4)$\n * If $x = 2$, $y = 0$ $\\rightarrow (2, 0)$\n * Draw a dashed straight line through these points.\n\n* **For $y = x + 1$:**\n * If $x = 0$, $y = 1$ $\\rightarrow (0, 1)$\n * If $x = 2$, $y = 3$ $\\rightarrow (2, 3)$\n * Draw a dashed straight line through these points.\n\n* **For $y = -1$:**\n * This is a horizontal dashed line passing through all points where the $y$-coordinate is $-1$."},{"type":"text","order":2,"title":"Use test point (0, 0)","content":"To determine which side of each boundary line to shade, we can test a coordinate that is not on any of the lines. A convenient choice is the origin, $(0, 0)$:\n\n1. **For $y < 4 - 2x$:**\n Substitute $(0,0)$:\n $$0 < 4 - 2(0) \\Rightarrow 0 < 4 \\quad \\text{(True)}$$\n Since this is **true**, $(0,0)$ lies in the wanted region for this line. We shade the side containing $(0,0)$ (the region below/to the left of $y = 4 - 2x$).\n\n2. **For $y > x + 1$:**\n Substitute $(0,0)$:\n $$0 > 0 + 1 \\Rightarrow 0 > 1 \\quad \\text{(False)}$$\n Since this is **false**, $(0,0)$ is not in the wanted region. We shade the opposite side of $(0,0)$ (the region above the line $y = x + 1$).\n\n3. **For $y > -1$:**\n Substitute $(0,0)$:\n $$0 > -1 \\quad \\text{(True)}$$\n Since this is **true**, $(0,0)$ lies in the wanted region. We shade the side containing $(0,0)$ (the region above the horizontal line $y = -1$)."},{"type":"text","order":3,"title":"Identify and label region R","content":"Now, we find the overlapping region that satisfies all three conditions:\n* Below the line $y = 4 - 2x$\n* Above the line $y = x + 1$\n* Above the line $y = -1$\n\nThe feasible region is **unbounded to the left**, not triangular. The lines $y = x + 1$ and $y = -1$ meet at $(-2,-1)$, while $y = x + 1$ and $y = 4 - 2x$ meet at $(1,2)$. The lower boundary is $y = -1$ to the left of $(-2,-1)$, then $y = x + 1$ up to $(1,2)$; the upper boundary is $y = 4 - 2x$. Shade this unbounded region and label it clearly with the letter **R** to secure all 3 marks."}]},{"id":"fce482a9-8595-4df9-bd98-690b15c99780:exclusion_shading","label":"Exclusion Shading (Shading Unwanted Regions)","steps":[{"type":"text","order":0,"title":"Understand the method","content":"To solve this problem using **Exclusion Shading**, we will use the unwanted regions to identify the required region:\n1. Find the boundary lines for each inequality and draw them as **dashed lines** (since all inequalities use strict inequality signs like $<$ and $>$ rather than $\\le$ or $\\ge$).\n2. For each inequality, identify the region that does **not** satisfy the inequality.\n3. The region left after excluding all unwanted regions is the target region. Shade this region and label it **R**, as required by the question."},{"type":"text","order":1,"title":"Plot first boundary line","content":"Let's start with the first inequality:\n$$y < 4 - 2x$$\n\nTo find the boundary line, we plot $$y = 4 - 2x$$.\n- When $x = 0$, $y = 4$. This gives the point $(0, 4)$.\n- When $x = 2$, $y = 0$. This gives the point $(2, 0)$.\n- When $x = -1$, $y = 6$. This gives the point $(-1, 6)$.\n\nDraw a **dashed line** passing through these points.\n\nSince we want the region where $y < 4 - 2x$, the **unwanted** region is above this line. Identify this unwanted region, but do not leave the required region unshaded: it will be shaded after all three inequalities have been considered.","highlights":[{"text":"y < 4 - 2x","location":"specification"}]},{"type":"text","order":2,"title":"Plot second boundary line","content":"Now look at the second inequality:\n$$y > x + 1$$\n\nTo find the boundary line, we plot $$y = x + 1$$.\n- When $x = 0$, $y = 1$. This gives the point $(0, 1)$.\n- When $x = 2$, $y = 3$. This gives the point $(2, 3)$.\n- When $x = -2$, $y = -1$. This gives the point $(-2, -1)$.\n\nDraw a **dashed line** passing through these points.\n\nSince we want the region where $y > x + 1$, the **unwanted** region is below this line. Identify this unwanted region, but do not leave the required region unshaded: it will be shaded after all three inequalities have been considered.","highlights":[{"text":"y > x + 1","location":"specification"}]},{"type":"text","order":3,"title":"Plot third boundary line","content":"Finally, consider the third inequality:\n$$y > -1$$\n\nTo find the boundary line, we plot the horizontal line:\n$$y = -1$$\n\nDraw this as a **dashed horizontal line** passing through $-1$ on the $y$-axis.\n\nSince we want the region where $y > -1$, the **unwanted** region is below the line $y = -1$. Identify this unwanted region, but do not leave the required region unshaded: it will be shaded after all three inequalities have been considered.","highlights":[{"text":"y > -1","location":"specification"}]},{"type":"text","order":4,"title":"Identify and label R","content":"After excluding the unwanted sides, identify the region that satisfies all three inequalities:\n- Above the dashed line $y = x + 1$\n- Above the dashed horizontal line $y = -1$\n- Below the dashed line $y = 4 - 2x$\n\nThis region is **unbounded to the left**, not triangular. Its lower boundary is $y = -1$ to the left of $(-2,-1)$, followed by $y = x + 1$ up to $(1,2)$. Its upper boundary is $y = 4 - 2x$.\n\nNow **shade this region** and write a clear capital letter **R** in it to complete the question.","highlights":[{"text":"Label the region R.","location":"specification"}]}]}],"generatedAt":"2026-06-12T22:14:03.034Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"075f5334-fa95-47bb-811d-09178a5493aa","revisionId":"3b9e7c09-92c0-44c6-bbd1-cb9aa8e4e67c","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"medium","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"f641cfbb-e6f6-4e32-8493-04f1ab86667a","label":"a","content":"On the grid shade the region that satisfies all these inequalities\n\n$$y < x + 4$$\n\n$$y > -x$$\n\n$$x < 3$$\n\nLabel the region **R**.\n\n![Coordinate grid](https://assets.mathsgenie.co.uk/figures/coords/4d33c73a04c0cb96b41aa7a902ef9134607eb5fce4e764c8392f7c32e67d7f27.png)","markscheme":"- Draws lines $y = x + 4$, $y = -x$, and $x = 3$ (all dashed). M1\n- Correct shading of the region bounded by the three lines. M1\n- Region correctly labeled **R**. A1\n\n![Solution](https://assets.mathsgenie.co.uk/figures/coords/11745287c9c52250c320d38a3ef6b032a516b5909b5d54cecd153d21c09b6781.png)","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Accurately draws the boundary lines $y = x + 4$, $y = -x$, and $x = 3$ as dashed lines.","code":"M"},{"id":"p2","marks":1,"description":"Correctly identifies and shades the region satisfying all three inequalities.","code":"M"},{"id":"p3","marks":1,"description":"Label the shaded region with the letter 'R'.","code":"A"}],"finalAnswers":["Region bounded by (-2, 2), (3, 7), and (3, -3)"],"generatedAt":"2026-05-06T14:01:11.937Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":{"methods":[{"id":"boundary_logic","label":"Interpreting Inequality Directions","steps":[{"type":"text","order":0,"title":"Identify Boundary Lines","content":"To represent the inequalities on a graph, we first identify the boundary lines by replacing the inequality symbols with equals signs:\n\n1. $$y = x + 4$$\n2. $$y = -x$$\n3. $$x = 3$$\n\nBecause all the inequalities ($<$ and $>$) are strict, we draw these boundary lines as **dashed lines** to indicate that the lines themselves are not part of the solution region.","highlights":[{"text":"y < x + 4","location":"specification"},{"text":"y > -x","location":"specification"},{"text":"x < 3","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=ineq1}{\\color{@sky}{y < x + 4}}$"},{"latex":"$$\\text{The symbol } \\color{@sky}{<} \\text{ means we shade } \\htmlData{anchor=dir1}{\\color{@sky}{\\text{below}}} \\text{ the line.}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"ineq1"},{"color":"sky","style":"text-color","anchor":"dir1"}],"connections":[{"to":"dir1","from":"ineq1","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Interpret the first inequality","description":"We determine which side of the line $y = x + 4$ to shade by looking at the inequality symbol."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=ineq2}{\\color{@amber}{y > -x}} \\quad \\text{and} \\quad \\htmlData{anchor=ineq3}{\\color{@purple}{x < 3}}$"},{"latex":"$$\\text{For } \\color{@amber}{>} \\text{, shade } \\htmlData{anchor=dir-above}{\\color{@amber}{\\text{above}}} \\text{ the line } y = -x$"},{"latex":"$$\\text{For } \\color{@purple}{<} \\text{, shade to the } \\htmlData{anchor=dir-\\left}{\\color{@purple}{\\text{left}}} \\text{ of } x = 3$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"ineq2"},{"color":"amber","style":"text-color","anchor":"dir-above"},{"color":"purple","style":"text-color","anchor":"ineq3"},{"color":"purple","style":"text-color","anchor":"dir-left"}],"connections":[{"to":"dir-above","from":"ineq2","color":"amber","label":null,"style":"solid"},{"to":"dir-left","from":"ineq3","color":"purple","label":null,"style":"solid"}]},"order":2,"title":"Interpret remaining inequalities","description":"We apply the same logic to the remaining two inequalities to find the required directions."},{"type":"text","order":3,"title":"Identify and Label Region R","content":"Now, we find the triangular region where all three conditions overlap:\n\n- Below the line $$y = x + 4$$\n- Above the line $$y = -x$$\n- To the left of the vertical line $$x = 3$$\n\nShade this central triangle and label it clearly with a capital **R**. This satisfies all three marks: drawing dashed lines, shading the correct intersection, and labeling the region.","highlights":[{"text":"Label the region R.","location":"specification"}]}]},{"id":"test_points","label":"Point Testing Method","steps":[{"type":"text","order":0,"title":"Draw the boundary lines","content":"First, we draw the equations of the lines as if they were equalities:\n\n1. $$y = x + 4$$ (a line with a y-intercept of $4$ and a gradient of $1$)\n2. $$y = -x$$ (a line passing through the origin with a gradient of $-1$)\n3. $$x = 3$$ (a vertical line passing through $3$ on the $x$-axis)\n\nBecause the inequalities are all strict ($<$ or $>$), we must use **dashed lines** to show that the points on the lines themselves are not included in the region.","calculator":[{"gdc":{"expression":"y < x + 4, y > -x, x < 3","instructions":"Go to the Graph menu. Change the relation type to inequality by pressing [TYPE] then [INEQ]. Enter the three inequalities."},"ti84":{"expression":"Y1 < X + 4, Y2 > -X","instructions":"Press [Y=]. Use the arrow keys to move to the left of 'Y1'. Press [ENTER] until the shading icon shows 'below' (for <) or 'above' (for >). Enter the equations."},"desmos":{"expression":"y < x + 4; y > -x; x < 3","instructions":"Type each inequality into a separate row. Desmos will automatically shade the overlapping region."},"description":"You can visualize the region by entering the inequalities directly into a graphing calculator."}],"highlights":[{"text":"y < x + 4","location":"specification"},{"text":"y > -x","location":"specification"},{"text":"x < 3","location":"specification"}]},{"type":"text","order":1,"title":"Choose a test point","content":"To find where the region $R$ lies, we pick a test point that is **not** on any of our boundary lines. A good choice is $(1, 1)$, as it is clearly not on the lines $y=x+4$, $y=-x$, or $x=3$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Test point: } (x, y) = (\\htmlData{anchor=pt-x}{\\color{@sky}{1}}, \\htmlData{anchor=pt-y}{\\color{@pink}{1}})$"},{"latex":"$$\\htmlData{anchor=sub-y1}{\\color{@pink}{1}} < \\htmlData{anchor=sub-x1}{\\color{@sky}{1}} + 4 \\Rightarrow 1 < 5 \\text{ (True)}$"},{"latex":"$$\\htmlData{anchor=sub-y2}{\\color{@pink}{1}} > -(\\htmlData{anchor=sub-x2}{\\color{@sky}{1}}) \\Rightarrow 1 > -1 \\text{ (True)}$"},{"latex":"$$\\htmlData{anchor=sub-x3}{\\color{@sky}{1}} < 3 \\text{ (True)}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"pt-x"},{"color":"pink","style":"text-color","anchor":"pt-y"}],"connections":[{"to":"sub-x1","from":"pt-x","color":"sky","label":null,"style":"solid"},{"to":"sub-y1","from":"pt-y","color":"pink","label":null,"style":"solid"}]},"order":2,"title":"Test the point","description":"We substitute our test point $(1, 1)$ into all three inequalities to see if they are all true."},{"type":"text","order":3,"title":"Shade and label region R","content":"Since our test point $(1, 1)$ satisfies all three inequalities, the region containing this point is the solution. Shade the triangular area bounded by the three dashed lines and label it clearly with a capital **R** to earn the final mark.","highlights":[{"text":"Label the region R.","location":"specification"}]}]},{"id":"exclusion_shading","label":"Elimination of Unwanted Regions","steps":[{"type":"text","order":0,"title":"Understand the method","content":"To find the region $R$, we will use the **Elimination of Unwanted Regions** method. Draw the boundary lines for each inequality, then identify and rule out the areas that **do not** satisfy them. Shade the region that remains, and label it $R$.\n\nNote that all three inequalities use strict symbols ($<$ or $>$), so all boundary lines will be **dashed** to show the line itself is not included in the region."},{"type":"text","order":1,"title":"Plot the line y = x + 4","content":"First, we draw the boundary line for $$y < x + 4$$. We can find two points to plot the line $$y = x + 4$$:\n- When $x = 0$, $y = 0 + 4 = 4$. Point: $(0, 4)$\n- When $y = 0$, $0 = x + 4 \\Rightarrow x = -4$. Point: $(-4, 0)$\n\nDraw a **dashed line** through these points. Since we want the region where $y$ is **less than** $x + 4$, rule out the area **above** the line. We will shade only the region that satisfies all three inequalities.","highlights":[{"text":"y < x + 4","location":"specification"}]},{"type":"text","order":2,"title":"Plot the line y = -x","content":"Next, look at $$y > -x$$. We plot the boundary line $$y = -x$$:\n- When $x = 0$, $y = 0$. Point: $(0, 0)$\n- When $x = 2$, $y = -2$. Point: $(2, -2)$\n\nDraw a **dashed line** through these points. We want the region where $y$ is **greater than** $-x$, so rule out the area **below** the line.","highlights":[{"text":"y > -x","location":"specification"}]},{"type":"text","order":3,"title":"Plot the line x = 3","content":"The final inequality is $$x < 3$$. The boundary line is $$x = 3$$, which is a **vertical line** passing through $3$ on the $x$-axis.\n\nDraw this as a **dashed line**. We want the region where $x$ is **less than** 3, so rule out the area to the **right** of the line.","highlights":[{"text":"x < 3","location":"specification"}]},{"type":"text","order":4,"title":"Label the region R","content":"After ruling out the three unwanted areas, the region satisfying all the inequalities is a triangle. Shade this triangle.\n\nIts vertices lie at:\n- $(-2, 2)$ where $y = x + 4$ and $y = -x$ intersect.\n- $(3, 7)$ where $y = x + 4$ and $x = 3$ intersect.\n- $(3, -3)$ where $y = -x$ and $x = 3$ intersect.\n\nThe dashed boundaries themselves are not included. Label the shaded region with a capital **R** to complete the task."}]}],"generatedAt":"2026-06-08T02:37:19.630Z","modelVersion":"gemini-3-flash-preview","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/review/270a9042-ccc4-420c-9499-93f7f515748c.mp4","durationSeconds":91,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/review/270a9042-ccc4-420c-9499-93f7f515748c.jpg"}},{"id":"af5b1546-a5e7-459c-a115-f3eef0d4b028","revisionId":"d5112089-62c5-4abd-9aa8-3267d80d375b","questionNumber":null,"questionType":"LA","specification":"![Coordinate plane showing a shaded region bounded by the y-axis, a horizontal line at y = -2, and a line with a negative gradient and a positive y-intercept.](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/af5b1546-a5e7-459c-a115-f3eef0d4b028/image-1-554c685dda5a053a3a9e324f.svg)","difficultyBand":"medium","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"8077de84-b1e2-43c5-a73e-399c0a76b50b","label":"a","content":"Write down the three inequalities that define the shaded region.","markscheme":"- $x \\geq 0$ B1\n- $y \\geq -2$ B1\n- $y \\leq -2x + 4$ B2","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"States the vertical boundary inequality $x \\geq 0$.","code":"B"},{"id":"p2","marks":1,"description":"States the horizontal boundary inequality $y \\geq -2$.","code":"B"},{"id":"p3","marks":2,"description":"States the inequality for the sloped line with an intercept $y \\leq -2x + 4$.","code":"B"}],"finalAnswers":["$$x \\geq 0$","$$y \\geq -2$","$$y \\leq -2x + 4$"],"generatedAt":"2026-05-06T14:01:43.910Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":{"methods":[{"id":"8077de84-b1e2-43c5-a73e-399c0a76b50b:slope_intercept_test_point","label":"Slope-Intercept and Test Point","steps":[{"type":"text","order":0,"title":"Identify boundary lines","content":"To define the shaded region, we need to find the equations of the three straight lines that form its boundaries:\n1. The vertical boundary on the left (the $y$-axis).\n2. The horizontal boundary at the bottom.\n3. The slanted boundary line."},{"type":"text","order":1,"title":"Find the vertical boundary","content":"The vertical boundary on the left is the $y$-axis, which has the equation:\n\n$$x = 0$$\n\nSince the shaded region is to the **right** of this line, the values of $x$ are greater than or equal to $0$.\n\nThis gives us our first inequality:\n\n$$x \\geq 0$$"},{"type":"text","order":2,"title":"Find the horizontal boundary","content":"The horizontal boundary at the bottom passes through $-2$ on the $y$-axis. Its equation is:\n\n$$y = -2$$\n\nSince the shaded region lies **above** this line, the values of $y$ must be greater than or equal to $-2$.\n\nThis gives us our second inequality:\n\n$$y \\geq -2$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"The slanted line crosses the $y$-axis at $(0, 4)$ and the $x$-axis at $(2, 0)$."},{"latex":"$$m = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{\\htmlData{anchor=y-diff}{\\color{@red}{0 - 4}}}{\\htmlData{anchor=x-diff}{\\color{@sky}{2 - 0}}} = \\htmlData{anchor=gradient}{\\color{@purple}{-2}}$"},{"latex":"$$y = \\htmlData{anchor=m-sub}{\\color{@purple}{-2}}x + \\htmlData{anchor=c-val}{\\color{@green}{4}}$"}],"highlights":[{"color":"red","style":"text-color","anchor":"y-diff"},{"color":"sky","style":"text-color","anchor":"x-diff"},{"color":"purple","style":"text-color","anchor":"gradient"},{"color":"green","style":"text-color","anchor":"c-val"}],"connections":[{"to":"m-sub","from":"gradient","color":"purple","label":"gradient (m)","style":"solid"}]},"order":3,"title":"Find the slanted line equation","description":"We use the slope-intercept form $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept ($4$)."},{"type":"text","order":4,"title":"Use a test point","content":"To find the direction of the inequality for the slanted line, pick a test point inside the shaded region, such as $(1, 0)$.\n\nSubstitute $x = 1$ and $y = 0$ into the expression $y$ and $-2x + 4$:\n\n$$\\begin{aligned} y &\\text{ vs } -2x + 4 \\\\ 0 &\\text{ vs } -2(1) + 4 \\\\ 0 &\\text{ vs } 2 \\end{aligned}$$\n\nSince $0 \\leq 2$, the inequality for this boundary is:\n\n$$y \\leq -2x + 4$$"},{"type":"text","order":5,"title":"State the final inequalities","content":"The three inequalities that define the shaded region are:\n\n1. $$x \\geq 0$$\n2. $$y \\geq -2$$\n3. $$y \\leq -2x + 4$$"}]},{"id":"8077de84-b1e2-43c5-a73e-399c0a76b50b:intercepts_and_visual_direction","label":"Axis Intercepts and Visual Direction","steps":[{"type":"text","order":0,"title":"Identify vertical boundary","content":"Let's first inspect the vertical boundary on the left side of the shaded region.\n\nThe shaded region is bounded on the left by the $y$-axis, which has the equation:\n\n$$x = 0$$\n\nSince the shaded region lies to the right of the $y$-axis (where $x$ values are positive), this boundary is represented by the inequality:\n\n$$x \\geq 0$$"},{"type":"text","order":1,"title":"Identify horizontal boundary","content":"Next, let's find the horizontal boundary at the bottom of the shaded region.\n\nLooking at the graph, there is a solid horizontal line passing through $-2$ on the $y$-axis. Its equation is:\n\n$$y = -2$$\n\nSince the shaded region lies above this line, the inequality is:\n\n$$y \\geq -2$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{y-intercept } (0, \\htmlData{anchor=c-val}{\\color{@sky}{4}}) \\implies c = \\color{@sky}{4}$"},{"latex":"$$m = \\frac{y_2 - y_1}{x_2 - x_1} = \\frac{\\color{@sky}{4} - 0}{0 - \\htmlData{anchor=x-val}{\\color{@amber}{2}}} = \\htmlData{anchor=m-val}{\\color{@purple}{-2}}$"},{"latex":"$$y = \\htmlData{anchor=m-eq}{\\color{@purple}{m}}x + \\htmlData{anchor=c-eq}{\\color{@sky}{c}} \\implies y = \\color{@purple}{-2}x + \\color{@sky}{4}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"c-val"},{"color":"amber","style":"text-color","anchor":"x-val"},{"color":"purple","style":"text-color","anchor":"m-val"}],"connections":[{"to":"c-eq","from":"c-val","color":"sky","label":null,"style":"solid"},{"to":"m-eq","from":"m-val","color":"purple","label":null,"style":"solid"}]},"order":2,"title":"Find slanted boundary equation","description":"Now, let's determine the equation of the slanted boundary line using its axis intercepts. The line crosses the $y$-axis at $(0, 4)$ and the $x$-axis at $(2, 0)$."},{"type":"text","order":3,"title":"Determine slanted inequality","content":"Now we determine the direction of the inequality for this slanted line:\n\n1. Visually, the shaded region lies **below** the boundary line $y = -2x + 4$.\n2. Because the boundary line is solid, we include the boundary itself.\n\nTherefore, the inequality is:\n\n$$y \\leq -2x + 4$$"},{"type":"text","order":4,"title":"State the final inequalities","content":"The three inequalities that define the shaded region are:\n\n1. $$x \\geq 0$$\n2. $$y \\geq -2$$\n3. $$y \\leq -2x + 4$$"}]}],"generatedAt":"2026-06-13T01:38:23.770Z","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 10

Inequalities on Graphs Questions

Inequalities on Graphs

Question 10

Inequalities on Graphs Questions