Finding Regions using Inequalities
How do we draw inequalities on a graph?
- First, see Straight Line Graphs (y = mx + c)
To graph an inequality;
- DRAW the line (as if using “=”) for each inequality
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Use a solid line for ≤ or ≥ (to indicate the line is included)
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Use a dotted line for < or > (to indicate the line is not included)
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- DECIDE which side of line is wanted.
- Below line if "y ≤ ..." or "y < ..."
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Above line if "y ≥ ..." or "y > ..."
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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
- 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
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(Graphing software often shades the area that is required but this is easily overcome by reversing the inequality sign)
Worked example
On the axes given below, show the region that satisfies the three inequalities;
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 "<"
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
For , shade the unwanted region to the right of . If unsure, check with a point
Finally, don't forget to label the region R