Questions: The region R satisfies the inequalities x ≥ 0, x+2 y ≤ 32 and two other inequalities. a) What are the other two inequalities? b) A new inequality changes region R so that it is now a kite. What is the new inequality?

The region R satisfies the inequalities x ≥ 0, x+2 y ≤ 32 and two other inequalities.
a) What are the other two inequalities?
b) A new inequality changes region R so that it is now a kite. What is the new inequality?

Transcript text: The region R satisfies the inequalities $x \geq 0, x+2 y \leq 32$ and two other inequalities. a) What are the other two inequalities? b) A new inequality changes region $R$ so that it is now a kite. What is the new inequality?

Solution

Solution Steps

Step 1: Find the first inequality

The orange line passes through the points (0,10) and (10,20). Its slope is (20-10)/(10-0) = 1. The equation of the line is _y_ = _x_ + 10. Since the region R is below the line, the inequality is _y_ ≤ _x_ + 10.

Step 2: Find the second inequality

The purple line passes through the points (0,16) and (30,0). Its slope is (16-0)/(0-30) = –16/30 = –8/15. The equation of the line is _y_ = –(8/15)_x_ + 16. Since the region R is below the line, the inequality is _y_ ≤ –(8/15)_x_ + 16.

Step 3: Find the inequality that would change the region to a kite

The new inequality is formed by changing the inequality sign of the orange line equation. If it changes from _y_ ≤ _x_ + 10 to _y_ ≥ _x_ + 10, the region R becomes a kite.