Given that X,Z are positive Y is negative and W can be either positive or zero or negative. The given conditions are : W4+X3+Y2+Z≤4 X3+Z≥2 W4+Y2≤2 Y2+Z≥3 For W4+Y2≤2 : Since Y is negative but Y2 is always positive and must be less than 2 because is a non negative value. HenceY=−1 is the only possibility. For W this can take any value among −1,0,1 For Y2+Z≥3 : Since Y=−1,Z must be at least equal to 2 so the value of Y2+Z≥3 is greater than 2.X is a positive value and must at least be equal to 1. The condition W2+X2+Y2 here has all the independent values: X2,Y2,Z2,W2 are non negative. For W4+X3+Y2+Z≤4 : Since the value of Z is at least equal to 2 the value of is equal to 1.Since X is a positive number in order to have the condition of W4+X3+Y2+Z≤4 satisfied. The value of Z must be the minimum possible so that X3+Y2+Z to have a value equal to 4 when X takes the minimum possible positive value equal to 1. Hence X must be 1. W must be equal to 0 so that : W4+X3+Y2+Z≤4= The sum =(0+1+1+2)=4. The only possible case- W2+X2+Y2+Z2 =(0+1+1+4) =6.