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CIE IGCSE Maths: Extended Your notes

Solving & Graphing Inequalities

Contents
Graphical Inequalities

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Graphical Inequalities
Your notes
Finding Regions using Inequalities
How do we draw inequalities on a graph?
First, see Linear Graphs
To graph an inequality;
1. DRAW the line (as if using “=”) for each inequality
Use a solid line for ≤ or ≥ (to indicate the line is included)
Use dotted line for < or > (to indicate the line is not included)

2. DECIDE which side of line is wanted.

Below line if "y ≤ ..." or "y < ..."
Above line if "y ≥ ..." or "y > ..."
Use a point that's not on the line as a test if unsure; substitute its x and y value into the inequality to
examine whether the inequality holds true on that side of the line

3. Shade UNWANTED side of each line (unless the question says otherwise)
This is because it is easier, with pen/ pencil/ paper at least, to see which region has not been
shaded than it is to look for a region that has been shaded 2-3 times or more
(Graphing software often shades the area that is required but this is easily overcome by reversing
the inequality sign)

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Worked example
Your notes
On the axes given below, show the region that satisfies the three inequalities;

3x + 2y ≥ 12 y < 2x x < 3
Label the region R.

First draw the three straight lines, , and , using your knowledge of
Straight Line Graphs (y = mx + c). You may wish to rearrange to the form
first:

The line takes a solid line because of the "≥" while the lines and take
dotted lines because of the "<"

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Your notes

Now we need to shade the unwanted regions

For (or ), the unwanted region is below the line. We can check this
with the point (0, 0);

is false therefore (0, 0) does lie in the unwanted region for

For , the unwanted region is above the line. If unsure, check with another point, for example (1,
0)

is true, so (1, 0) lies in the wanted (i.e. unshaded) region for

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For , shade the unwanted region to the right of . If unsure, check with a point

Your notes

Finally, don't forget to label the region R

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Interpreting Graphical Inequalities

How do we interpret inequalities on a graph? Your notes
First, see Linear Graphs
To interpret inequalities/ to find a region defined by inequalities;
1. Write down the EQUATION of each line on the graph
2. REMEMBER that lines are drawn with:
A solid line for ≤ or ≥ (to indicate line included in region)
A dotted line for < or > (to indicate line not included)
3. REPLACE = sign with:
≤ or < if shading below line

≥ or > if shading above line

(Use a point to test if not sure)

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Worked example
Your notes
Write down the three inequalities which define the shaded region on the axes below.

First, using your knowledge of Straight Line Graphs (y = mx + c), define the three lines as equations,
ignoring inequality signs;

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Your notes

Now decide which inequality signs to use

For , the shaded region is above the line, and the line is dotted, so the inequality is

Check by substituting a point within the shaded region into this inequality. For example, using (2, 4) as
marked on the graph above;

" " is true, so the inequality is correct

For , the shaded region is below the line, and the line is solid, so the inequality is

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or
Your notes
Again, check by substituting (2, 4) into the inequality;

" " is true, so the inequality is correct

For , the shaded region is to the right of the solid line so the inequality is

(Vertical and horizontal inequality lines can also be checked by substituting in a point if you are unsure)

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