To determine the correct graph that models the inequality [tex]\( y \leq -\frac{2}{5} x + 2 \)[/tex], let's break down the key components of the inequality and how they should be represented graphically:
1. Identify the y-intercept:
The y-intercept is the point where the line crosses the y-axis. From [tex]\( y \leq -\frac{2}{5} x + 2 \)[/tex], the y-intercept is 2. This means the line will cross the y-axis at the point (0, 2).
2. Determine the slope:
The slope of the line is given by the coefficient of [tex]\( x \)[/tex], which is [tex]\(-\frac{2}{5}\)[/tex]. This indicates that for every 5 units you move to the right along the x-axis, you move down 2 units. Thus, the line has a negative slope and will decline from left to right.
3. Graph the line:
Start by plotting the y-intercept at (0, 2). Then, using the slope [tex]\(-\frac{2}{5}\)[/tex], plot another point. From (0, 2), move 5 units to the right to reach (5, 0) and then move down 2 units to reach the point (5, 0-2) = (5, -2). Draw a straight line through these points.
4. Shading the appropriate region:
Since the inequality is [tex]\( y \leq -\frac{2}{5} x + 2 \)[/tex], we are interested in the region below the line (including the line itself). This region includes all points where the y-value is less than or equal to what the line defines.
Summarizing these characteristics:
- Y-Intercept: The line crosses the y-axis at (0, 2).
- Slope: The line declines with a slope of [tex]\(-\frac{2}{5}\)[/tex].
- Shading: The region below and including the line should be shaded.
So, to model the inequality [tex]\( y \leq -\frac{2}{5} x + 2 \)[/tex], look for a graph showing:
- A line crossing the y-axis at (0, 2).
- The line has a downward slope of [tex]\(-\frac{2}{5}\)[/tex], where the line falls 2 units for every 5 units it moves to the right.
- The shaded area is below the line, indicating the region where [tex]\( y \)[/tex] is less than or equal to the line's value.
The graph that best fits all these criteria is the correct one.
