graph the equation 2x + 3y = 6. Now graph the ineqaulity 2x + 3y is less than or equal to 6, by shading all the points (x, y) that fit it. Notice that this means shading all the points on one side of the line you drew. Which sides? Use a test point like (0, 0) to decide.

In the graph above:

1. The blue line represents the equation \(2x + 3y = 6\), which is the boundary between the two regions.
2. The shaded region below the line shows the inequality \(2x + 3y \leq 6\). All points in this region satisfy the inequality.
3. We use the point \((0, 0)\) to test which side of the line should be shaded. Substituting \((0, 0)\) into the inequality gives:
   \[
   2(0) + 3(0) = 0 \leq 6
   \]
   Since this is true, the point \((0, 0)\) satisfies the inequality, confirming that the region below the line (including the line itself) should be shaded.

Would you like any further details on this, or do you have any questions?

Here are five related questions to explore further:
1. What happens if we change the inequality to \(2x + 3y < 6\)?
2. How would you graph \(2x + 3y > 6\)?
3. How does the slope of the line affect the shading of the inequality?
4. How can we verify if a random point \((x, y)\) lies within the shaded region?
5. What is the effect of changing the constant in the equation, for example \(2x + 3y = 8\)?

**Tip**: Always choose simple test points like \((0, 0)\) when available, as they make verifying inequalities easier.

Graph the equation 2x + 3y = 6. Now graph the inequality 2x + 3y is less than or equal to 6, by shading all the points (x, y) that fit it. Notice that this means shading all the points on one side of the line you drew. Which sides? Use a test point like (0, 0) to decide.

Explore how to graph the linear equation 2x + 3y = 6 and shade the region for the inequality 2x + 3y ≤ 6. Learn how to use test points like (0, 0) to determine which side of the line to shade. Suitable for students in Grades 8-10.

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