Question

y < -\frac{5}{6}x + 3
plot points on the boundary line. select the line to switch between solid and dotted. select a region to shade it.

Explanation:

Step1: Identify the boundary line equation

The inequality is \( y < -\frac{5}{6}x + 3 \). The boundary line is \( y = -\frac{5}{6}x + 3 \) (dotted line since the inequality is strict, \( < \)).

Step2: Find two points on the boundary line

• When \( x = 0 \): \( y = -\frac{5}{6}(0) + 3 = 3 \). So the point is \( (0, 3) \).
• When \( x = 6 \): \( y = -\frac{5}{6}(6) + 3 = -5 + 3 = -2 \). So the point is \( (6, -2) \).

Step3: Draw the boundary line

Plot the points \( (0, 3) \) and \( (6, -2) \), then draw a dotted line through them (since \( y < \dots \), not \( \leq \)).

Step4: Determine the region to shade

To find which region to shade, test a point not on the line, e.g., \( (0, 0) \).
Substitute into the inequality: \( 0 < -\frac{5}{6}(0) + 3 \) → \( 0 < 3 \), which is true. So shade the region containing \( (0, 0) \) (below the dotted line \( y = -\frac{5}{6}x + 3 \)).

Answer:

1. Plot the boundary line (dotted) through \( (0, 3) \) and \( (6, -2) \).
2. Shade the region below the dotted line (containing \( (0, 0) \)).