Shade the region represented by the inequations. `x+yle3` and `3x-2yge4`.

Answer

To solve the problem of shading the region represented by the inequalities \( x + y \leq 3 \) and \( 3x - 2y \geq 4 \), we will follow these steps:

### Step 1: Rewrite the Inequalities
We have two inequalities:
1. \( x + y \leq 3 \)
2. \( 3x - 2y \geq 4 \)

### Step 2: Convert Inequalities to Equations
To find the boundary lines, we convert the inequalities into equations:
1. \( x + y = 3 \)
2. \( 3x - 2y = 4 \)

### Step 3: Find Intercepts for the First Equation
For \( x + y = 3 \):
- When \( x = 0 \), \( y = 3 \) (Point: \( (0, 3) \))
- When \( y = 0 \), \( x = 3 \) (Point: \( (3, 0) \))

### Step 4: Plot the First Line
Plot the points \( (0, 3) \) and \( (3, 0) \) on the graph and draw a solid line through these points since the inequality is less than or equal to (≤).

### Step 5: Determine the Shaded Region for the First Inequality
To determine which side of the line to shade, we can test a point. Let's use the origin \( (0, 0) \):
- Substitute into the inequality: \( 0 + 0 \leq 3 \) (True)
Thus, we shade the region below the line \( x + y = 3 \).

### Step 6: Find Intercepts for the Second Equation
For \( 3x - 2y = 4 \):
- When \( x = 0 \), \( -2y = 4 \) → \( y = -2 \) (Point: \( (0, -2) \))
- When \( y = 0 \), \( 3x = 4 \) → \( x = \frac{4}{3} \) (Point: \( \left(\frac{4}{3}, 0\right) \))

### Step 7: Plot the Second Line
Plot the points \( (0, -2) \) and \( \left(\frac{4}{3}, 0\right) \) on the graph and draw a solid line through these points since the inequality is greater than or equal to (≥).

### Step 8: Determine the Shaded Region for the Second Inequality
Again, we can test the origin \( (0, 0) \):
- Substitute into the inequality: \( 3(0) - 2(0) \geq 4 \) (False)
Thus, we shade the region above the line \( 3x - 2y = 4 \).

### Step 9: Identify the Common Shaded Region
Now, we look for the common shaded region between the two inequalities. The region below the line \( x + y = 3 \) and above the line \( 3x - 2y = 4 \) is our solution.

### Final Step: Shade the Common Region
Shade the area that satisfies both inequalities on the graph.

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