How do you determine that an inequality represents the area above the line and when do you include the line in the solution?
Suppose that the inequality is
3y < x + 3.
I would first plot the graph of the line
3y = x + 3.
This line divides the points in the plane into 3 regions:
The points on the line are the points that satisfy the equation 3y = x + 3. The other two regions are the points that satisfy 3y < x + 3 and 3y > x + 3. The question is: which is which?
- the points on the line
- the points above the line, and
- the points below the line
To answer this question I choose some point in one of the two regions, say (2,0) which is below the line.
Now check to see which inequality is satisfied by (2,0). When x = 2 any y = 0 the left side is left side is 0 and the right side is 5 and 0 < 5, so
(2,0) satisfies 3y < x + 3.
Since (2,0) is below the line, the points that satisfy 3y < x + 3 are the points below the line.
If the inequality is
then this represents two expressions
3y < x + 3
3y = x + 3
The points that satisfy are the points that satisfy either 3y < x + 3 or 3y = x + 3. That is the points that are either below the line or on the line. Thus the line is included in the solution if the inequality is or .
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