To graph the inequality [tex]\( y \leq 2x - 4 \)[/tex], follow these steps to determine the boundary line:
### Step-by-Step Solution
1. Identify the equation of the boundary line:
The boundary line for the inequality [tex]\( y \leq 2x - 4 \)[/tex] is given by the equation:
[tex]\[
y = 2x - 4
\][/tex]
This is a linear equation in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
2. Determine the slope and y-intercept:
- The slope [tex]\( m \)[/tex] is 2.
- The y-intercept [tex]\( b \)[/tex] is -4.
3. Plot the y-intercept on the graph:
Start by plotting the point (0, -4) on the graph, which is the point where the line crosses the y-axis.
4. Use the slope to find another point on the line:
Since the slope [tex]\( m = 2 \)[/tex], it means for every unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 2 units. Starting from the y-intercept (0, -4):
- Increase [tex]\( x \)[/tex] by 1 to get [tex]\( x = 1 \)[/tex].
- Increase [tex]\( y \)[/tex] by 2 to get [tex]\( y = -4 + 2 = -2 \)[/tex].
Thus, another point on the line is (1, -2).
5. Draw the boundary line:
Plot the points (0, -4) and (1, -2) on the graph. Draw a straight line through these points. Since the inequality symbol is [tex]\( \leq \)[/tex] (less than or equal to), the boundary line itself should be solid, indicating that points on the line satisfy the inequality.
6. Understand the inequality:
The inequality [tex]\( y \leq 2x - 4 \)[/tex] means that we also shade the region below this boundary line, as it includes all points where [tex]\( y \)[/tex] is less than or equal to [tex]\( 2x - 4 \)[/tex].
### Identifying the Correct Graph
When you look at the options, you need to find the graph where:
- The boundary line [tex]\( y = 2x - 4 \)[/tex] is correctly drawn with the y-intercept at (0, -4).
- The line is solid, indicating that the boundary itself is part of the solution.
- The shaded region is below the boundary line.
Carefully review the options and select the one that meets all of these criteria. Identify the graph option that matches the details explained here.
(Please note, since physical graph options A, B, C, etc. are not provided here, refer to your actual test options to select the correct graph).
