Question

The region satisfying the inequalities \(y-x\geq 2, x+y\leq 5, x\geq 0\) and \(y\geq 0\) is

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

Draw the lines \(y=x+2\) and \(y=5-x\) and find the region on the correct sides.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
\(y-x\ge2\) means on or above the line \(y=x+2\). \(x+y\le5\) means on or below \(y=5-x\). \(x\ge0,y\ge0\) keep the region in the first quadrant.

Step 2: Key Formula or Approach
Lines meet where \(x+2=5-x\): \(x=\tfrac32\), \(y=\tfrac72\). The line \(y=x+2\) cuts the y-axis at 2 and \(y=5-x\) cuts it at 5.

Step 3: Detailed Explanation
The region is a triangle with vertices \((0,2)\), \((0,5)\) and \(\left(\tfrac32,\tfrac72\right)\): bounded on the left by the y-axis, below by the rising line and above by the falling line.
The figure in option (A) shows this small triangle shaded against the y-axis between the two lines, ending where the lines meet.

Final Answer:
Figure (A) shows the required region. \[ \boxed{\text{(A)}} \]