A region is defined by the inequalities x + y > 6 and x − y > −4.
Consider the three statements:
1 x > 1
2 y > 5
3 (x+y)(x−y) > −24
Which of the above statements is/are true for every point in the region?
- Anone
- B1 only
- C2 only
- D3 only
- E1 and 2 only
- F1 and 3 only
- G2 and 3 only
- H1, 2 and 3
Show the answer and worked solution
answer · B
- Anone
- B1 only
- C2 only
- D3 only
- E1 and 2 only
- F1 and 3 only
- G2 and 3 only
- H1, 2 and 3
Adding the two inequalities gives 2x > 2, so statement 1 always holds. Statement 2 fails at (10, 0): there x+y = 10 > 6 and x − y = 10 > −4, yet y is not greater than 5. Statement 3 is the trap: multiplying x+y > 6 by x−y > −4 is not legitimate because x−y can be negative. Take x + y = 100 and x − y = −3.9 (that is x = 48.05, y = 51.95), which lies in the region; the product is −390, far below −24. So only statement 1.