Question

1. what graph best represents the solution for $y > 3x - 4$?

Explanation:

Step1: Analyze the inequality type

The inequality is \( y > 3x - 4 \). First, consider the boundary line. The equation of the boundary line is \( y = 3x - 4 \), which is a linear equation in slope - intercept form (\( y=mx + b \), where \( m = 3 \) (slope) and \( b=- 4 \) (y - intercept)). Since the inequality is \( y>3x - 4 \) (not \( y\geq3x - 4 \)), the boundary line should be a dashed line (because the points on the line \( y = 3x-4 \) are not included in the solution set).

Step2: Determine the region to shade

To determine which side of the line to shade, we can use a test point. A common test point is the origin \((0,0)\) (as long as the line does not pass through it). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y>3x - 4 \):
\( 0>3(0)-4 \)
\( 0>- 4 \), which is a true statement. So, we shade the region that contains the origin \((0,0)\) (the region above the line \( y = 3x - 4 \) since the slope is positive and the y - intercept is - 4).

So, the graph of \( y>3x - 4 \) should have a dashed line with the equation \( y = 3x - 4 \) (slope of 3, y - intercept at (0, - 4)) and the region above the line shaded.

(Note: Since the problem asks for the graph, and we can't draw it here, but the key features are a dashed line \( y = 3x-4 \) and shading above the line.)

Answer:

The graph should have a dashed line \( y = 3x - 4 \) (with slope \( 3 \) and y - intercept at \( (0,-4) \)) and the region above the line shaded.