The graphical solution of linear inequalities `x+y ge 5 " and " x -y le 3, `

Options

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Explanation

To solve the given linear inequalities graphically, we will follow these steps:

### Step 1: Understand the inequalities
We have two inequalities:
1. \( x + y \geq 5 \)
2. \( x - y \leq 3 \)

### Step 2: Convert inequalities to equations
To graph these inequalities, we first convert them into equations:
1. \( x + y = 5 \)
2. \( x - y = 3 \)

### Step 3: Find intercepts for the first equation
For the equation \( x + y = 5 \):
- When \( x = 0 \), \( y = 5 \) (Point: \( (0, 5) \))
- When \( y = 0 \), \( x = 5 \) (Point: \( (5, 0) \))

### Step 4: Plot the first line
Plot the points \( (0, 5) \) and \( (5, 0) \) on the coordinate system and draw a straight line through these points. Since the inequality is \( \geq \), we shade the area above this line.

### Step 5: Find intercepts for the second equation
For the equation \( x - y = 3 \):
- When \( x = 0 \), \( y = -3 \) (Point: \( (0, -3) \))
- When \( y = 0 \), \( x = 3 \) (Point: \( (3, 0) \))

### Step 6: Plot the second line
Plot the points \( (0, -3) \) and \( (3, 0) \) on the coordinate system and draw a straight line through these points. Since the inequality is \( \leq \), we shade the area below this line.

### Step 7: Identify the feasible region
The feasible region is where the shaded areas from both inequalities overlap. This region represents all the solutions that satisfy both inequalities.

### Step 8: Conclusion
The graphical solution to the inequalities \( x + y \geq 5 \) and \( x - y \leq 3 \) is the shaded area where both conditions are satisfied.

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