Questions: Graph the inequality on the axes below. y ≤ 3x-6

Transcript text: Graph the inequality on the axes below. \[ y \leq 3 x-6 \]

Solution

Solution Steps

Step 1: Determine two points on the line

To graph the line y = 3x - 6, we need to find two points that satisfy this equation. We can choose any x-values and substitute them into the equation to get the corresponding y-values.

When x = 0, y = 3(0) - 6 = -6. So, the first point is (0, -6). When x = 2, y = 3(2) - 6 = 0. So, the second point is (2, 0).

Step 2: Plot the points and draw the line

Plot the two points (0, -6) and (2, 0) on the coordinate plane. Since the inequality includes the equal sign (≤), we draw a solid line through these two points.

Step 3: Shade the region representing the inequality

The inequality is y ≤ 3x - 6. To determine which side of the line to shade, we can pick a test point that is not on the line. A convenient test point is (0, 0). Substituting the coordinates of this point into the inequality, we get 0 ≤ 3(0) - 6, which simplifies to 0 ≤ -6. This is false. Therefore, the region containing (0, 0) is not the solution. Shade the other side of the line, the region that does not contain (0,0).

Final Answer:

The graph of the inequality y ≤ 3x - 6 is the solid line passing through the points (0,-6) and (2,0), and the region below the line is shaded.

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