An amusement park ride features two types of cars: one that accommodates 6 passengers and another that accommodates 3 passengers. If the ride designer intends to ensure that each run carries at least 12 passengers, which graph depicts the possible combinations of 6-passenger cars (x) and 3-passenger cars (y)?

The problem can be modeled as a linear inequality that represents the combinations of 6-passenger cars (x) and 3-passenger cars (y) that meet the designer's requirement. The inequality that represents the scenario where at least 12 passengers are carried in each run is:

6x + 3y ≥ 12

To find the graph that depicts this situation, we need to identify the region on a coordinate plane that satisfies this inequality.

First, we can rewrite the inequality in terms of y:

3y ≥ 12 - 6x y ≥ 4 - 2x

Now we can graph this as a line on a coordinate plane, with x as the number of 6-passenger cars and y as the number of 3-passenger cars. To find the line, we can identify the intercepts:

- When x = 0 (no 6-passenger cars), y = 4. This gives us the y-intercept at (0, 4). - When y = 0 (no 3-passenger cars), x = 2. This gives us the x-intercept at (2, 0).

Connect these points to draw the line y = 4 - 2x on the graph. Since we are looking for combinations that include at least 12 passengers, we want the region of the graph where the values of y and x satisfy the inequality 6x + 3y ≥ 12. This will be the shaded region above and including the line.

Remember that in a real-world scenario in an amusement park, the number of cars can only be whole numbers (integers). Therefore the solutions of the inequality on the graph (the possible combinations of the number of cars) will be represented by the points with integer coordinates that lie within the shaded region above the line.

Extra: In graphing linear inequalities, the first step is usually to graph the line as if the inequality sign were an equal sign. In this case, you graph the line y = 4 - 2x. The type of line used to represent the graph can indicate whether the line itself is included in the solutions:

- A solid line is used when the inequality includes the equal sign (≥ or ≤), meaning that points on the line are solutions to the inequality. - A dashed line is used when the inequality does not include the equal sign (> or <), indicating that points on the line are not solutions to the inequality.

In this scenario, we used a solid line because points on the line y = 4 - 2x are valid solutions (since they also carry at least 12 passengers).

After graphing the line, the next step is to determine which side of the line to shade. To do this, you can pick a test point that is not on the line (often the origin, (0,0), is used) and substitute its coordinates into the inequality. If the inequality is true with the test point, the region that includes that point is shaded. If it's false, the opposite side is shaded.

Since amusement park rides typically have a fixed number of cars, the points we consider on the graph should be whole numbers (integers), as fractional cars are not practical in this context. These integer points represent the only viable solutions for the number of each type of car.