Inequations `3x- y ge 3 and 4x-y ge 4`

Options

A. have solution for positive x and y
B. have no solution for positive x and y
C. have solution for all x
D. have solution for all y

Answer

To solve the given inequalities \(3x - y \geq 3\) and \(4x - y \geq 4\), we will follow these steps:

### Step 1: Rewrite the inequalities in slope-intercept form
We start by rewriting each inequality in the form \(y \leq mx + b\).

1. For the first inequality:
\[ 3x - y \geq 3 \implies -y \geq -3x + 3 \implies y \leq 3 - 3x \]

2. For the second inequality:
\[ 4x - y \geq 4 \implies -y \geq -4x + 4 \implies y \leq 4 - 4x \]

### Step 2: Find the intercepts
Next, we find the x-intercepts and y-intercepts for both lines.

1. For the line \(y = 3 - 3x\):
- **x-intercept**: Set \(y = 0\):
\[ 0 = 3 - 3x \implies 3x = 3 \implies x = 1 \]
- **y-intercept**: Set \(x = 0\):
\[ y = 3 - 3(0) = 3 \]

2. For the line \(y = 4 - 4x\):
- **x-intercept**: Set \(y = 0\):
\[ 0 = 4 - 4x \implies 4x = 4 \implies x = 1 \]
- **y-intercept**: Set \(x = 0\):
\[ y = 4 - 4(0) = 4 \]

### Step 3: Plot the lines on a graph
Now we can plot the lines based on the intercepts we found:

- The line \(y = 3 - 3x\) passes through points (1, 0) and (0, 3).
- The line \(y = 4 - 4x\) passes through points (1, 0) and (0, 4).

### Step 4: Determine the feasible region
Since both inequalities are in the form \(y \leq mx + b\), we shade the region below each line.

- For \(y \leq 3 - 3x\), shade below the line.
- For \(y \leq 4 - 4x\), shade below the line.

### Step 5: Identify the common feasible area
The common feasible area is where the shaded regions overlap. This area represents the solutions to the system of inequalities.

### Step 6: Analyze the options
Now we analyze the options based on the feasible region:

- The feasible region is in the first quadrant where both \(x\) and \(y\) are positive.
- Therefore, the correct option is that there are solutions for positive \(x\) and \(y\).

### Final Answer
The correct option is that there are solutions for positive \(x\) and \(y\).

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