Graphical Inequalities Revision Notes for Edexcel GCSE Maths

Graphical inequalities

Understanding graphical inequalities involves learning how to represent mathematical relationships on a coordinate plane by shading specific regions. While this might seem challenging at first, following a systematic approach makes the process straightforward and manageable.

The four-step method

When working with graphical inequalities, you'll always need to shade a region on a graph. The key is to follow a clear, methodical approach that ensures accuracy every time.

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The four-step method provides a systematic approach that works for any graphical inequality problem, regardless of complexity. Following these steps in order will help you avoid common mistakes and ensure consistent results.

Step 1: Convert inequalities to equations

The first step requires transforming each inequality into an equation. This means replacing the inequality symbol ( $\<$ , $>$ , $\leq$ , or $\geq$ ) with an equals sign ( $=$ ). For example, if you have the inequality $y \< 3x + 2$ , you would change it to $y = 3x + 2$ .

This conversion allows you to draw the boundary line that separates the regions where the inequality is true from where it's false. Every inequality has a corresponding boundary line that acts as the dividing line between the solution region and the non-solution region.

Step 2: Draw the appropriate line type

Once you have your equations, you need to draw the boundary lines on your graph. The type of line you draw depends on the original inequality symbol:

For strict inequalities ( $\<$ or $>$ ), draw a dotted line. This indicates that points on the line itself are not included in the solution set. The boundary exists but isn't part of the answer.

For non-strict inequalities ( $\leq$ or $\geq$ ), draw a solid line. This shows that points on the line are included in the solution set. The boundary is part of the solution region.

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Line Type Rules:

$\<$ or $>$ → dotted line (boundary not included)
$\geq$ or $\leq$ → solid line (boundary included)

This is a critical concept that directly affects your final answer. Using the wrong line type will cost you marks even if everything else is correct.

Remember that the line type is crucial for indicating whether the boundary is included in your final answer.

Step 3: Determine which side to shade

This step involves testing points to find out which side of each boundary line satisfies the original inequality. The most common approach is to test the origin $(0, 0)$ by substituting these coordinates into the original inequality.

If substituting $(0, 0)$ makes the inequality true, then the origin is on the correct side of the line, and you should shade the region containing the origin. If the substitution makes the inequality false, then shade the opposite side of the line.

When the origin lies directly on the boundary line, you'll need to choose a different test point. Pick coordinates that are easy to work with, such as $(1, 0)$ or $(0, 1)$ , and apply the same testing process.

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Why use the origin? The origin $(0, 0)$ is the preferred test point because it makes calculations simple - you're often just comparing zero to another number. This reduces the chance of arithmetic errors and speeds up your work.

Step 4: Shade the final region

After determining the correct side for each inequality, you need to identify the region that satisfies all the given inequalities simultaneously. This region is where all the individual solution areas overlap.

The final shaded region represents all the coordinate points that make every inequality in the system true at the same time. This overlapping area is your complete solution.

Working through an example

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Worked Example: Finding the Region

Consider finding the region that satisfies $x + y \< 5$ , $y \leq x + 2$ , and $y > 1$ simultaneously.

Step 1: Convert each inequality to an equation: $x + y = 5$ , $y = x + 2$ , and $y = 1$ .

Step 2: Draw the appropriate lines: dotted lines for $x + y = 5$ and $y = 1$ (since these use $\<$ and $>$ ), and a solid line for $y = x + 2$ (since this uses $\leq$ ).

Step 3: Test the origin $(0, 0)$ in each original inequality:

For $x + y \< 5$ : $0 + 0 \< 5$ ✓ (true - origin is on correct side)
For $y \leq x + 2$ : $0 \leq 0 + 2$ ✓ (true - origin is on correct side)
For $y > 1$ : $0 > 1$ ✗ (false - origin is on wrong side)

Step 4: The solution region is where all three conditions are met: below the line $x + y = 5$ , below or on the line $y = x + 2$ , and above the line $y = 1$ .

Important exam considerations

Always read the question carefully, as you might be asked to label the region instead of shading it. Pay attention to whether the question asks for shading or labelling, and respond accordingly.

After finding your solution region, it's wise to pick a point inside the shaded area and verify that it satisfies all the original inequalities. This serves as a useful check of your work.

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Common Exam Pitfall: Never throw away marks by misreading the question requirements. Take time to understand exactly what's being asked before beginning your solution. Some questions ask for labelling (e.g., "Label the region R"), while others ask for shading.

Remember!

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Key Points to Remember:

Convert inequalities to equations by replacing inequality signs with equals signs
Use dotted lines for strict inequalities ( $\<$ or $>$ ) and solid lines for non-strict inequalities ( $\leq$ or $\geq$ )
Test the origin $(0, 0)$ to determine which side of each line to shade, unless it lies on the boundary
The final solution is the region where all individual inequality solutions overlap
Always double-check your work by testing a point in your final shaded region

Graphical Inequalities (Edexcel GCSE Maths): Revision Notes