Find the solutions set of inequalities ` 0 le x le 5, 0 le 2y le7`

Answer

To find the solution set of the inequalities \( 0 \leq x \leq 5 \) and \( 0 \leq 2y \leq 7 \), we will follow these steps:

### Step 1: Understand the inequalities
The inequalities can be broken down as follows:
1. \( 0 \leq x \leq 5 \) indicates that \( x \) can take any value from 0 to 5, inclusive.
2. \( 0 \leq 2y \leq 7 \) can be simplified by dividing the entire inequality by 2, giving us \( 0 \leq y \leq \frac{7}{2} \).

### Step 2: Graph the inequalities
To visualize the solution set, we will graph the inequalities on a coordinate plane.

1. **Graph \( x = 0 \) and \( x = 5 \)**:
- The line \( x = 0 \) is the y-axis.
- The line \( x = 5 \) is a vertical line parallel to the y-axis at \( x = 5 \).

2. **Graph \( y = 0 \) and \( y = \frac{7}{2} \)**:
- The line \( y = 0 \) is the x-axis.
- The line \( y = \frac{7}{2} \) is a horizontal line parallel to the x-axis at \( y = 3.5 \).

### Step 3: Identify the feasible region
The feasible region is where all the inequalities overlap:
- The region where \( x \) is between 0 and 5 (including the boundaries).
- The region where \( y \) is between 0 and \( \frac{7}{2} \) (including the boundaries).

### Step 4: Determine the vertices of the feasible region
The vertices of the rectangle formed by the intersection of these inequalities are:
1. \( (0, 0) \)
2. \( (5, 0) \)
3. \( (5, \frac{7}{2}) \)
4. \( (0, \frac{7}{2}) \)

### Step 5: Conclusion
The solution set of the inequalities is the rectangular area defined by the vertices mentioned above, including the boundaries. Therefore, the solution set can be expressed as:
\[ \{ (x, y) \mid 0 \leq x \leq 5 \text{ and } 0 \leq y \leq \frac{7}{2} \} \]