(a) \( \left\{\begin{array}{l}x+y \leq 50 \\ A=(20,40) \\ B=(10,30)\end{array}\right. \)

It looks like you have a system of inequalities and some points defined in a two-dimensional space. Let's break down the information you provided: 1. **Inequality**: \( x + y \leq 50 \) 2. **Points**: - Point \( A = (20, 40) \) - Point \( B = (10, 30) \) ### Step 1: Graph the Inequality The inequality \( x + y \leq 50 \) can be represented graphically. To do this, we first find the boundary line by setting the equation \( x + y = 50 \). - **Intercepts**: - When \( x = 0 \), \( y = 50 \) (point \( (0, 50) \)) - When \( y = 0 \), \( x = 50 \) (point \( (50, 0) \)) Now, we can draw the line connecting these two points. The area below this line (including the line itself) represents the solution to the inequality \( x + y \leq 50 \). ### Step 2: Check the Points Next, we can check whether the points \( A \) and \( B \) lie within the feasible region defined by the inequality. - **Point A = (20, 40)**: \[ 20 + 40 = 60 \quad (\text{not } \leq 50) \] Point A does not satisfy the inequality. - **Point B = (10, 30)**: \[ 10 + 30 = 40 \quad (\text{is } \leq 50) \] Point B satisfies the inequality. ### Conclusion - The point \( A = (20, 40) \) does not lie within the feasible region defined by the inequality \( x + y \leq 50 \). - The point \( B = (10, 30) \) lies within the feasible region. If you have any further questions or need additional analysis, feel free to ask!