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; for \( y = 3x-4 \), when \( x = 0 \), \( y=-4
eq0 \), so the origin is not on the line).
Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y>3x - 4 \):
\( 0>3(0)-4 \)
\( 0>- 4 \), which is a true statement.
This means that the origin \((0,0)\) lies in the solution region. So we shade the region above the dashed line \( y = 3x - 4 \) (since the origin is above the line \( y = 3x - 4 \) when \( x = 0 \), and the inequality is \( y>3x - 4 \)).

Step3: Describe the graph features

• The boundary line: Dashed line with slope \( 3 \) and y - intercept \( - 4 \) (passes through \((0,-4)\) and we can find another point using the slope: from \((0,-4)\), moving up 3 units and right 1 unit gives \((1,-1)\)).
• The shaded region: The region above the dashed line \( y = 3x - 4 \).

(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 \( 3 \), y - intercept \( - 4 \)) and the region above this dashed line shaded.