Question

graph the inequality.
$y < 2x - 4$

Explanation:

Step1: Graph the boundary line

The inequality is \( y < 2x - 4 \). First, graph the line \( y = 2x - 4 \). This is a linear equation in slope - intercept form (\( y=mx + b \)) where the slope \( m = 2 \) and the y - intercept \( b=-4 \). Since the inequality is \( y < 2x - 4 \) (not \( y\leq2x - 4 \)), we use a dashed line to represent the boundary line (because the points on the line \( y = 2x-4 \) are not included in the solution set of \( y < 2x - 4 \)).

To graph the line, start by plotting the y - intercept. The y - intercept is at \( (0,-4) \). Then, use the slope to find another point. The slope \( m = 2=\frac{2}{1} \), which means from the point \( (0,-4) \), we can move up 2 units and to the right 1 unit to get the point \( (1,-2) \). We can also move down 2 units and to the left 1 unit from \( (0,-4) \) to get \( (-1,-6) \). Draw a dashed line through these points.

Step2: Determine the region to shade

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 \( (0,0) \) (since it is not on the line \( y = 2x-4 \)). Substitute \( x = 0 \) and \( y = 0 \) into the inequality \( y<2x - 4 \):

\( 0<2(0)-4 \)

\( 0 < - 4 \)

This statement is false. So, the point \( (0,0) \) is not in the solution set. Therefore, we shade the region that does not contain the point \( (0,0) \). In other words, we shade the region below the dashed line \( y = 2x-4 \).

(Note: Since the problem asks to graph the inequality, the final answer is the graph with the dashed line \( y = 2x - 4 \) and the region below the line shaded. But since we are providing a textual description of the graphing process, the key steps are as above.)

Answer:

The graph of \( y < 2x-4 \) consists of a dashed line with slope 2 and y - intercept - 4 (passing through points like \( (0,-4) \), \( (1,-2) \), \( (-1,-6) \)) and the region below this dashed line is shaded.