Question

The shaded region is the solution set of the inequalities

• A. 5x + 4y ≥ 20, x ≤ 6, y ≥ 3, x ≥ 0, y ≥ 0
• B. 5x + 4y ≤ 20, x ≤ 6, y ≤ 3, x ≥ 0, y ≥ 0
• C. 5x + 4y ≥ 20, x ≤ 6, y ≤ 3, x ≥ 0, y ≥ 0
• D. 5x + 4y ≥ 20, x ≥ 6, y ≤ 3, x ≥ 0, y ≥ 0

Answer 1

The Correct Option is C

The inequalities represent a system of constraints defining the shaded region in the graph.
1. \( 5x + 4y \geq 20 \) represents a line that cuts the region into two parts. We are interested in the region above this line.
2. \( x \leq 6 \) represents a vertical line at \( x = 6 \) and restricts the region to the left of this line.
3. \( y \leq 3 \) represents a horizontal line at \( y = 3 \) and restricts the region to below this line.
4. \( x \geq 0 \) and \( y \geq 0 \) ensure that the region is in the first quadrant. Thus, the solution set of the inequalities that defines the shaded region is option (C).

The correct answer is (C) : 5x + 4y ≥ 20, x ≤ 6, y ≤ 3, x ≥ 0, y ≥ 0.

Answer 2

Given: The shaded region is bounded by a set of linear inequalities. We are to identify the correct set.

Step 1: Consider the line: 

\(5x + 4y = 20\)

Find intercepts:

• For x-intercept: set \(y = 0\) ⇒ \(5x = 20 \Rightarrow x = 4\)
• For y-intercept: set \(x = 0\) ⇒ \(4y = 20 \Rightarrow y = 5\)

This gives us points \((4, 0)\) and \((0, 5)\), and the line is solid because the inequality involves ≥ or ≤.

Step 2: Determine the region for \(5x + 4y \geq 20\):

Pick a test point below the line, say \((0,0)\):

\(5(0) + 4(0) = 0 < 20\), which does not satisfy the inequality, so the region is above the line.

Step 3: Additional constraints:

• \(x \leq 6\): left of vertical line \(x = 6\)
• \(y \leq 3\): below horizontal line \(y = 3\)
• \(x \geq 0\), \(y \geq 0\): in the first quadrant

Conclusion: The correct system of inequalities describing the shaded region is:

\(5x + 4y \geq 20,\ x \leq 6,\ y \leq 3,\ x \geq 0,\ y \geq 0\)

Answer: Option 3