Graphical Inequalities Revision Notes for AQA GCSE Maths

Graphical inequalities

When working with graphical inequalities, you'll need to shade regions on a coordinate grid that satisfy given conditions. This might seem complex at first, but following a systematic approach makes it much more manageable.

Understanding the basics

Graphical inequalities involve finding areas on a graph where certain mathematical conditions are met. Instead of just plotting single lines or points, you're identifying entire regions that satisfy inequality statements like $x + y < 5$ or $y > 2x + 1$ .

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The key difference between regular graphing and graphical inequalities is that instead of drawing individual lines or points, you're working with entire regions or areas that satisfy your conditions.

The four-step method

There's a reliable four-step method you can follow every time you encounter these problems:

Step 1: Convert inequalities to equations

Take each inequality and replace the inequality symbol (like $<$ , $>$ , $\leq$ , or $\geq$ ) with an equals sign. This gives you the boundary lines for your regions.

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Examples of conversion:

$x + y < 5$ becomes $x + y = 5$
$y \leq x + 2$ becomes $y = x + 2$
$y > 1$ becomes $y = 1$

Step 2: Draw the boundary lines

Now you need to sketch these equations on your coordinate grid. The type of line you draw depends on the original inequality symbol:

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Line Types:

Dotted lines are used for strict inequalities ( $<$ or $>$ ). These show that points on the line itself don't satisfy the inequality.
Solid lines are used for inclusive inequalities ( $\leq$ or $\geq$ ). These show that points on the line do satisfy the inequality.

Step 3: Determine which side of each line you need

This is often the trickiest part. You need to work out which side of each boundary line contains the points that satisfy your original inequality. The most effective way to do this is by testing a point.

The origin (0, 0) is usually the best point to test, as long as it doesn't lie on any of your boundary lines. Simply substitute the coordinates into your original inequality and see if it makes the statement true or false.

For instance, if you're working with $x + y < 5$ and test the origin:

$0 + 0 < 5$ gives you $0 < 5$ , which is true
This means the origin is on the correct side of the line

If testing the origin gives you a false statement, then you want the opposite side of the line.

Step 4: Shade the final region

The region you're looking for is where all your inequalities are satisfied simultaneously. This means finding the area where all the correct sides of your boundary lines overlap.

Working through an example

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Worked Example: Finding the Region

Let's say you need to find the region that satisfies all three of these inequalities:

$x + y < 5$
$y \leq x + 2$
$y > 1$

Step 1: Convert to equations and draw boundary lines

$x + y = 5$ (dotted line, since it's a strict inequality)
$y = x + 2$ (solid line, since it includes equality)
$y = 1$ (dotted line, since it's a strict inequality)

Step 2: Test the origin (0, 0) in each inequality

For $x + y < 5$ : $0 + 0 < 5$ is true, so the origin is on the correct side
For $y \leq x + 2$ : $0 \leq 0 + 2$ is true, so the origin is on the correct side
For $y > 1$ : $0 > 1$ is false, so the origin is on the wrong side

Step 3: Determine regions This tells you that you want:

The region below the line $x + y = 5$
The region below or on the line $y = x + 2$
The region above the line $y = 1$

Step 4: Find the overlap The final shaded area is where all three of these conditions are met at the same time.

Important tips to remember

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Critical Points to Remember:

Always read the question carefully. Sometimes you might be asked to label the region instead of shading it, or you might need to identify specific points rather than areas.

Once you've found your region, it's good practice to pick a point inside it and verify that this point satisfies all the original inequalities. This gives you confidence that you've got the right answer.

Remember that dotted lines mean the boundary itself isn't included, while solid lines mean it is included. This distinction can be crucial for getting full marks.

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Key Points to Remember:

Convert inequalities to equations first to find your boundary lines
Use dotted lines for $<$ and $>$ , solid lines for $\leq$ and $\geq$
Test a point (usually the origin) to determine which side of each line you need
Shade the overlap where all conditions are satisfied simultaneously
Double-check your answer by testing a point in your final region

Graphical Inequalities (AQA GCSE Maths): Revision Notes