The solution set of the linear inequalities ` 2x + y ge 8` and ` x + 2y ge 10 ` is

Options

A.
B.
C.
D. None of these

Explanation

To solve the linear inequalities \( 2x + y \geq 8 \) and \( x + 2y \geq 10 \), we will follow these steps:

### Step 1: Convert inequalities to equations
We start by converting the inequalities into equations to find the boundary lines.

1. For the first inequality \( 2x + y = 8 \).
2. For the second inequality \( x + 2y = 10 \).

### Step 2: Find intercepts for the first equation
To graph the line \( 2x + y = 8 \), we can find the x-intercept and y-intercept.

- **X-intercept**: Set \( y = 0 \):
\[ 2x + 0 = 8 \implies x = 4 \]
So, the x-intercept is \( (4, 0) \).

- **Y-intercept**: Set \( x = 0 \):
\[ 2(0) + y = 8 \implies y = 8 \]
So, the y-intercept is \( (0, 8) \).

### Step 3: Find intercepts for the second equation
Next, we find the intercepts for the line \( x + 2y = 10 \).

- **X-intercept**: Set \( y = 0 \):
\[ x + 2(0) = 10 \implies x = 10 \]
So, the x-intercept is \( (10, 0) \).

- **Y-intercept**: Set \( x = 0 \):
\[ 0 + 2y = 10 \implies y = 5 \]
So, the y-intercept is \( (0, 5) \).

### Step 4: Graph the lines
Now we will graph the two lines using the intercepts we found:

1. Draw the line through the points \( (4, 0) \) and \( (0, 8) \) for the equation \( 2x + y = 8 \).
2. Draw the line through the points \( (10, 0) \) and \( (0, 5) \) for the equation \( x + 2y = 10 \).

### Step 5: Determine the feasible region
To find the solution set for the inequalities, we need to identify which side of the lines satisfies the inequalities.

1. For \( 2x + y \geq 8 \):
- Test the origin \( (0, 0) \):
\[ 2(0) + 0 \geq 8 \implies 0 \geq 8 \quad \text{(False)} \]
- Since the origin does not satisfy the inequality, the feasible region is above the line \( 2x + y = 8 \).

2. For \( x + 2y \geq 10 \):
- Test the origin \( (0, 0) \):
\[ 0 + 2(0) \geq 10 \implies 0 \geq 10 \quad \text{(False)} \]
- Since the origin does not satisfy this inequality either, the feasible region is above the line \( x + 2y = 10 \).

### Step 6: Identify the intersection of the regions
The solution set is the region that is above both lines. This region is unbounded and extends infinitely in the direction where both inequalities are satisfied.

### Final Solution
The solution set of the linear inequalities \( 2x + y \geq 8 \) and \( x + 2y \geq 10 \) is the region above both lines in the coordinate plane.

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