Question

graph the linear inequality.
y < 3x - 4

Explanation:

Step1: Graph the boundary line

The inequality is \( y < 3x - 4 \). First, graph the line \( y = 3x - 4 \). This is a linear equation in slope - intercept form (\( y=mx + b \), where \( m = 3 \) (slope) and \( b=-4 \) (y - intercept)). Plot the y - intercept at \( (0,-4) \). Then, use the slope \( m = 3=\frac{3}{1} \) to find another point. From \( (0,-4) \), move up 3 units and right 1 unit to get the point \( (1,-1) \). Since the inequality is \( y < 3x - 4 \) (not \( y\leq3x - 4 \)), the boundary line should be a dashed line (to indicate that the points on the line are not included in the solution set).

Step2: Shade the solution region

To determine which side of the line to shade, we can use a test point that is not on the line. A common test point is \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y<3x - 4 \):
\( 0<3(0)-4 \)
\( 0 < - 4 \), which is false. So, the point \( (0,0) \) is not in the solution set. We shade the side of the line that does not contain the point \( (0,0) \).

Answer:

The graph consists of a dashed line with slope 3 and y - intercept - 4, and the region below the line is shaded.