Question

graph the inequality.
( y < 2x - 3 )

Explanation:

Step1: Graph the boundary line

The inequality is \( y < 2x - 3 \). First, we consider the boundary line \( y = 2x - 3 \). This is a linear equation in slope - intercept form (\( y=mx + b \), where \( m = 2 \) is the slope and \( b=-3 \) is the y - intercept). Since the inequality is strict (\( < \)), we draw a dashed line for \( y = 2x - 3 \). To graph the line, we start at the y - intercept \( (0,-3) \). Then, using the slope \( m = 2=\frac{2}{1} \), we move up 2 units and right 1 unit from the y - intercept to get another point, for example, \( (1,-1) \), and draw the dashed line through these points.

Step2: Determine the region to shade

We need to find which side of the line \( y = 2x - 3 \) satisfies the inequality \( y<2x - 3 \). We can use a test point that is not on the line. A common test point is the origin \( (0,0) \). Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \( 0<2(0)-3\)? \( 0 < - 3\) is false. So the origin is not in the solution region. We shade the region that does not contain the origin. Since the slope of the line is positive, the region below the dashed line (because the test point above the line did not satisfy the inequality) should be shaded.

Answer:

1. Draw a dashed line for \( y = 2x-3 \) (passing through \( (0, - 3) \) and \( (1,-1) \) etc.).
2. Shade the region below the dashed line (the region that does not include the origin \( (0,0) \) when tested).