Question

graph this inequality:
$y \geq \frac{7}{6}x - 4$
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 type

The inequality is \( y \geq \frac{7}{6}x - 4 \). Since the inequality is "greater than or equal to", the boundary line \( y=\frac{7}{6}x - 4 \) should be a solid line (because the inequality includes equality, so the points on the line are part of the solution set).

Step2: Find two points on the boundary line

To graph the line \( y=\frac{7}{6}x - 4 \), we can find the x - intercept and y - intercept or other points.

• Y - intercept: When \( x = 0 \), \( y=\frac{7}{6}(0)-4=-4 \). So one point is \( (0, - 4) \).
• Another point: Let's choose \( x = 6 \) (to make the fraction calculation easier). Then \( y=\frac{7}{6}(6)-4=7 - 4 = 3 \). So another point is \( (6,3) \).

Step3: Determine the region to shade

To determine which side of the line to shade, we can test a point not on the line. A common test point is the origin \( (0,0) \).
Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y\geq\frac{7}{6}x - 4 \):
\( 0\geq\frac{7}{6}(0)-4 \)
\( 0\geq - 4 \), which is true. So we shade the region that includes the origin (the region above the line, since the origin is above the line \( y=\frac{7}{6}x - 4 \) when \( x = 0 \)).

(Note: Since this is a graphing problem, the final answer is the graph with a solid line through \( (0,-4) \) and \( (6,3) \) (and other points on \( y = \frac{7}{6}x-4 \)) and the region above the line shaded.)

Answer:

1. Draw a solid line for \( y=\frac{7}{6}x - 4 \) (passing through points like \( (0, - 4) \) and \( (6,3) \)).
2. Shade the region above the solid line (the region that includes the origin \( (0,0) \) when tested in the inequality).