Which inequality represents all points above the line \(y = 3\)?
Above means larger y-values.
Points above the line have y-values greater than 3, so \(y > 3\).
Inequalities on Graphs
Graphical inequalities show regions that satisfy certain conditions. In GCSE Maths, this involves shading areas and identifying solution sets.
Overview
An inequality on a graph shows all the points that make an inequality true.
Instead of just drawing a line, you must also show the correct region.
\( y \geq 2x + 1 \)
The boundary line represents where the inequality is equal, and the shaded region shows all solutions.
What you should understand after this topic
- Understand what the boundary line represents
- Know when to use a solid or dashed line
- Decide which side of the line to shade
- Test a point such as (0,0) to check the region
- Read and interpret a shaded region on a graph
Key Definitions
Inequality
A statement using symbols such as \( < \), \( > \), \( \leq \) or \( \geq \).
Boundary Line
The line you get by replacing the inequality sign with an equals sign.
Region
The shaded part of the graph showing all solutions.
Solid Line
Used when the boundary is included, for \( \leq \) or \( \geq \).
Dashed Line
Used when the boundary is not included, for \( < \) or \( > \).
Test Point
A point used to check which side of the boundary should be shaded.
Key Rules
\( \leq \) or \( \geq \)
Use a <strong>solid</strong> boundary line.
\( < \) or \( > \)
Use a <strong>dashed</strong> boundary line.
Boundary line
Replace the sign with \( = \) first.
Shading
Use a test point to choose the correct side.
How to Solve
Step 1: Draw the boundary line
Replace the inequality with an equation to find the boundary. Draw the line using methods from linear graphs.
\( y < 2x + 1 \quad \Rightarrow \quad y = 2x + 1 \)
Draw the line \( y = 2x + 1 \) first.
Exam tip: Always draw the boundary line before shading.
Step 2: Solid or dashed line
Include boundary
\( \leq \) or \( \geq \) → solid line
Exclude boundary
\( < \) or \( > \) → dashed line
Step 3: Decide where to shade
There are two ways to decide which side to shade.
Exam tip: Use the quick rule, then confirm with a test point if unsure.
Method 1: Test point
Try a point like \((0,0)\). If it satisfies the inequality, shade that side.
Method 2: Quick rule
For \(y >\) → shade above
For \(y <\) → shade below
Step 4: x and y inequalities
\( x > 2 \) → shade right
\( x < 2 \) → shade left
y inequalities
Shade above or below the line.
x inequalities
Shade left or right of the line.
Step 5: Reading inequalities from graphs
To work backwards from a graph, use your knowledge of linear graphs.
Exam thinking: Solid line means include the boundary.
- Find the equation of the boundary line.
- Check if the line is solid or dashed.
- Decide if the region is above, below, left or right.
- Choose the correct inequality sign.
Step 6: Horizontal and vertical lines
See linear graphs for drawing lines.
Horizontal line
\( y \geq 4 \) → solid line at 4, shade above
Vertical line
\( x < 2 \) → dashed line at 2, shade left
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on interpreting regions, drawing boundaries, and identifying solutions on graphs.
Edexcel
On the grid, draw the line \( y = x + 2 \). Shade the region where \( y > x + 2 \).
Edexcel
Draw the line \( y = 2x - 1 \). Shade the region satisfying \( y \leq 2x - 1 \).
Edexcel
Write down the inequality represented by the shaded region bounded by the line \( y = -x + 4 \), where the region is below the line and includes the boundary.
Edexcel
A point \( (2,5) \) is tested in the inequality \( y > x + 1 \). Determine whether the point is in the solution region.
Edexcel
Explain why a dashed line is used for some inequalities on graphs.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on shading regions, interpreting boundaries, and checking whether points satisfy inequalities.
AQA
Represent the inequality \( y < -2x + 3 \) on a graph.
AQA
Represent the inequality \( y \geq 3 \) on a graph.
AQA
The region satisfies both \( y \geq x - 2 \) and \( y < 4 \). Draw and shade the solution region.
AQA
A region is shaded above the line \( y = 2x + 1 \), and the line is solid. Write down the inequality.
AQA
A point has coordinates \( (-1,2) \). Check whether it satisfies \( y \leq x + 4 \).
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising combined regions, reasoning, and writing inequalities from shaded graphs.
OCR
Draw and shade the region that satisfies \( y > 1 \) and \( y \leq -x + 5 \).
OCR
Write down two inequalities that describe a region above the x-axis and below the line \( y = x + 3 \).
OCR
A region is bounded by \( x \geq 0 \), \( y \geq 0 \), and \( y < -x + 6 \). Describe this region.
OCR
State whether the point \( (4,1) \) is a solution of the region defined by \( y > -x + 3 \).
OCR
Explain the difference between using a solid boundary line and a dashed boundary line when drawing inequalities.
Exam Checklist
Step 1
Change the inequality into an equation to get the boundary line.
Step 2
Choose solid or dashed correctly.
Step 3
Use a test point if you are unsure which side to shade.
Step 4
Check whether the region should be above, below, left or right.
Most common exam mistakes
Boundary line
Forgetting to replace the sign with \( = \) first.
Line type
Using the wrong line style for included or excluded boundaries.
Shading
Shading the wrong side of the line.
Reading graphs
Writing \( \leq \) instead of \( < \), or the other way round.
Common Mistakes
These are common mistakes students make when working with inequalities on graphs in GCSE Maths.
Using a solid line for strict inequalities
Incorrect
A student draws a solid line for \(<\) or \(>\).
Correct
Strict inequalities use a dashed line. Solid lines are only used when the boundary is included, such as \(\leq\) or \(\geq\).
Using a dashed line for inclusive inequalities
Incorrect
A student draws a dashed line for \(\leq\) or \(\geq\).
Correct
Inclusive inequalities require a solid line because the boundary is part of the solution.
Shading the wrong region
Incorrect
A student shades the opposite side of the line.
Correct
Test a point (usually \((0,0)\)) in the inequality. If it works, shade that side; if not, shade the other side.
Not drawing the boundary line first
Incorrect
A student tries to sketch the inequality directly.
Correct
Always start by drawing the boundary line as an equation (replace the inequality sign with \(=\)), then decide on shading.
Confusing x- and y-inequalities
Incorrect
A student treats \(x < 3\) the same as \(y < 3\).
Correct
\(x\)-inequalities produce vertical lines, while \(y\)-inequalities produce horizontal lines. Make sure you identify the correct orientation.
Misinterpreting regions for negative gradients
Incorrect
A student shades above or below incorrectly when the line slopes downwards.
Correct
For lines with negative gradients, use a test point to decide the correct region instead of relying on visual guessing.
Try It Yourself
Practise solving inequalities and representing them on graphs.
Foundation
Foundation Practice
Understand inequality regions and boundary lines.
Which inequality represents all points below the line \(y = 5\)?
Below means smaller y-values.
Points below the line have y-values less than 5, so \(y < 5\).
Which inequality includes the boundary line \(y = 2\)?
Inclusive inequalities use ≤ or ≥.
\(y \geq 2\) includes the line because it means greater than or equal to.
Write an inequality for all points on or below the line \(y = 4\).
Use ≤ when including the line.
On or below means \(y \leq 4\).
Which inequality represents points to the right of \(x = 2\)?
Right means larger x-values.
Points to the right have x-values greater than 2, so \(x > 2\).
Write an inequality for all points to the left of \(x = 7\).
Left means smaller x-values.
Points to the left have x-values less than 7, so \(x < 7\).
A dashed line is used on a graph. What does this mean?
Think about strict inequalities.
A dashed line means the boundary is not included (>, <).
Which symbol is used when the boundary line is included?
Think about 'equal to'.
We use ≤ or ≥ when the boundary is included.
Which inequality would use a solid line?
Solid lines include equality.
\(y \leq 6\) includes the boundary, so it uses a solid line.
Write an inequality for all points above or on the line \(y = 1\).
Above means greater than.
Above or on means \(y \geq 1\).
Higher
Higher Practice
Interpret and solve inequalities on coordinate graphs.
Which inequality represents the region above the line \(y = 2x + 1\)?
Above means y is greater.
Points above the line have y-values greater than \(2x + 1\).
Write an inequality for points below the line \(y = 3x - 2\).
Below means smaller y-values.
Points below the line satisfy \(y < 3x - 2\).
Which inequality includes the line \(y = -x + 4\)?
Look for ≥ or ≤.
\(y \geq -x + 4\) includes the boundary line.
Write an inequality for points on or below \(y = x - 3\).
Use ≤ for 'on or below'.
The inequality is \(y \leq x - 3\).
A point (2, 6) lies above the line \(y = 2x + 1\). Which inequality is correct?
Substitute the point into the equation.
Substitute: \(6 > 2(2) + 1 = 5\), so \(y > 2x + 1\).
Check if the point (1, 2) satisfies \(y > x + 1\). Write 'yes' or 'no'.
Substitute x = 1, y = 2.
Check: \(2 > 1 + 1\) → \(2 > 2\) is false. So the answer is no.
Which region represents \(x > 3\)?
Think horizontally.
x-values increase to the right, so the region is to the right.
Write an inequality for points above the line \(y = -2x + 5\).
Above means greater.
The inequality is \(y > -2x + 5\).
A student draws a solid line for \(y > 2x\). What is wrong?
Think about strict inequalities.
Since the inequality is strict (>), the line should be dashed.
Check if the point (0, 1) satisfies \(y \geq x + 1\). Write 'yes' or 'no'.
Substitute x = 0, y = 1.
Check: \(1 \geq 0 + 1\) → \(1 \geq 1\) is true. So yes.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What do inequality graphs show?
Regions where solutions satisfy an inequality.
What is a solid vs dashed line?
Solid means included (≤ or ≥), dashed means not included (< or >).
How do I find the correct region?
Test a point to see which side satisfies the inequality.