The set of points `(x,y)` satisfying the inequalities `x+yle1,-x-yle1` lies in the region bounded by the two straight lines passing through the respective pair of points

Options

A. `{(1,0)(0,1)}` and `{(-1,0),(0,-1)}`
B. `{(1,0),(1,1)}` and `{(-1,0),(0,-1)}`
C. `{(-1,0),(0,-1)}` and `{(1,0),(-1,1)}`
D. `{(1,0),(1,1)}` and `{(-1,0),(0,1)}`

Answer

To solve the problem of finding the set of points \((x,y)\) that satisfy the inequalities \(x + y \leq 1\) and \(-x - y \leq 1\), we will follow these steps:

### Step 1: Rewrite the inequalities
The inequalities can be rewritten as:
1. \(x + y \leq 1\)
2. \(-x - y \leq 1\) or equivalently \(x + y \geq -1\)

### Step 2: Graph the lines
Next, we will graph the lines corresponding to these inequalities.

1. **For the line \(x + y = 1\)**:
- To find the intercepts, set \(x = 0\) to find the y-intercept:
\[ 0 + y = 1 \implies y = 1 \quad \text{(point (0,1))} \]
- Set \(y = 0\) to find the x-intercept:
\[ x + 0 = 1 \implies x = 1 \quad \text{(point (1,0))} \]
- Plot the points (0,1) and (1,0) and draw the line through these points.

2. **For the line \(x + y = -1\)**:
- Set \(x = 0\) to find the y-intercept:
\[ 0 + y = -1 \implies y = -1 \quad \text{(point (0,-1))} \]
- Set \(y = 0\) to find the x-intercept:
\[ x + 0 = -1 \implies x = -1 \quad \text{(point (-1,0))} \]
- Plot the points (0,-1) and (-1,0) and draw the line through these points.

### Step 3: Determine the feasible region
Now we need to determine the region that satisfies both inequalities.

1. For \(x + y \leq 1\):
- The region is below the line \(x + y = 1\). We can test the origin (0,0):
\[ 0 + 0 \leq 1 \quad \text{(True)} \]
- Thus, the region below this line is valid.

2. For \(x + y \geq -1\):
- The region is above the line \(x + y = -1\). Testing the origin (0,0):
\[ 0 + 0 \geq -1 \quad \text{(True)} \]
- Thus, the region above this line is valid.

### Step 4: Identify the bounded region
The feasible region is the area that is both below the line \(x + y = 1\) and above the line \(x + y = -1\).

### Step 5: Identify the vertices of the bounded region
The vertices of the bounded region can be found at the intersection points of the lines:
- The line \(x + y = 1\) intersects with \(x + y = -1\) at the points (0,1) and (1,0) for the first line and (0,-1) and (-1,0) for the second line.

### Conclusion
The set of points \((x,y)\) satisfying the inequalities lies in the region bounded by the points (0,1), (1,0), (0,-1), and (-1,0).