The graph shows a system of inequalities with a shaded region. Let's break down the graph and analyze it step-by-step:

1. The solid line in the graph represents the boundary for one of the inequalities. The line has a slope of 2 and crosses the origin (0, 0), so the equation of the line is: $y = 2x$ Since the shading is below the line, the inequality associated with this line is: $y \leq 2x$

2. There is a horizontal dashed line at $y = -1$, and the shaded region is below this line, so the second inequality is: $y < -1$

So the system of inequalities is: $y \leq 2x \quad \text{and} \quad y < -1$

The correct answer is the third option: $y \leq 2x$ and $y < -1$.

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How do you determine the inequality signs from a graph?
2. What changes in the system if the lines were dashed instead of solid?
3. How would the system change if the shaded region were above the line?
4. How do you graph a system of inequalities by hand?
5. What happens if two inequality regions overlap?

Tip: Always look at the direction of the shading to decide if the inequality is "greater than" or "less than."