Question

graph this inequality:
3x + y ≥ 6
plot points on the boundary line. select the line to switch between solid and dotted. select a region to shade it.

Explanation:

Step1: Rewrite the inequality

Rewrite \( 3x + y \geq 6 \) in slope - intercept form (\( y=mx + b \)) by solving for \( y \). We get \( y\geq - 3x + 6 \).

Step2: Graph the boundary line

The boundary line is \( y=-3x + 6 \). This is a linear equation. To find two points on the line:

• When \( x = 0 \), \( y=-3(0)+6=6 \), so the point is \( (0,6) \).
• When \( x = 2 \), \( y=-3(2)+6 = 0 \), so the point is \( (2,0) \).

Since the inequality is \( \geq \), the boundary line should be solid (because the inequality includes equality). Plot the points \( (0,6) \) and \( (2,0) \) and draw a solid line through them.

Step3: Determine the region to shade

To determine which side of the line to shade, we can test a point not on the line. Let's use the origin \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( 3x + y\geq6 \): \( 3(0)+0=0\geq6 \) is false. So we shade the region that does not include the origin. That is, the region above the line \( y = - 3x+6 \).

(Note: Since the problem asks to graph and the final answer here is about the steps to graph, if we were to describe the graph, the boundary line is solid, passes through (0,6) and (2,0), and the region above the line is shaded.)

Answer:

1. Rewrite the inequality as \( y\geq - 3x + 6 \).
2. Graph the solid line \( y=-3x + 6 \) (using points like \( (0,6) \) and \( (2,0) \)).
3. Shade the region above the line (since \( (0,0) \) does not satisfy the inequality, we shade the side opposite to the origin relative to the line).