A linear inequality divides the plane into 3 parts: the line, and the regions on either side of the line. In the graph above, the line can be represented by

\[2x-3y=-6 \, \text{ or } \, y=\frac{2}{3} \normalsize x+2\]

How do we decide between \(\leq\), <, >, \(\geq\)?

One student decided that \(2x-3y>-6\) because the shading is above the line. How can we tell if she was correct?

Go to Desmos, graph the line \(2x – 3y = 6\)

Choose a test point in each of the regions.

Choose a point in the plane. For example, \((-3,\,4)\)

Is the point in the solution region?

Yes, so when I substitute in \(2x-3y>-6\), my resulting inequality should be true:

\(2x-3y>-6\) ? for \((-3,\,4)\)

Try a few test points in each region and a point on the line to verify.

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