Sure, let's go through the steps to graph the linear inequality [tex]\( y < 2x - 4 \)[/tex] step by step.
### Step 1: Graph the Boundary Line
First, we'll graph the boundary line for the inequality.
1. Equation of the boundary line: [tex]\( y = 2x - 4 \)[/tex].
2. Plotting points:
- Find the y-intercept: When [tex]\( x = 0 \)[/tex], [tex]\( y = 2(0) - 4 = -4 \)[/tex]. So, the y-intercept is (0, -4).
- Find another point: When [tex]\( x = 2 \)[/tex], [tex]\( y = 2(2) - 4 = 4 - 4 = 0 \)[/tex]. So, another point is (2, 0).
3. Draw the line: Connect these points with a straight line. Since the inequality is [tex]\( y < 2x - 4 \)[/tex], and not [tex]\( y \leq 2x - 4 \)[/tex], we draw a dashed line to indicate that points on the line are not included in the solution set.
### Step 2: Shade the Appropriate Region
To determine which side of the line to shade, we need to test a point that is not on the line. A common test point is the origin (0,0), as long as it is not on the line itself.
1. Test the point (0, 0):
- Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality: [tex]\( 0 < 2(0) - 4 \)[/tex].
- Simplify: [tex]\( 0 < -4 \)[/tex], which is false.
Since the test point (0,0) does not satisfy the inequality, we shade the opposite side of the line. In this case, we shade the region below the line [tex]\( y = 2x - 4 \)[/tex].
### Summary of the Steps:
1. Draw a dashed line for [tex]\( y = 2x - 4 \)[/tex].
2. Identify and shade the region below the line, as this represents [tex]\( y < 2x - 4 \)[/tex].
### Visualization:
Here is a basic sketch of what the graph should look like:
1. Draw the y-axis and x-axis.
2. Plot the points (0, -4) and (2, 0).
3. Draw a dashed line passing through these points.
4. Shade the region below the dashed line, indicating the solution set for [tex]\( y < 2x - 4 \)[/tex].
The graph visually represents all the [tex]\( (x, y) \)[/tex] pairs that satisfy the inequality [tex]\( y < 2x - 4 \)[/tex].
