Question

Which statement about the following system of inequalities is true?

y- 1.75 (x-5) less than or equal to 0
2y - 2x^2 + 16 less than or equal to 0

The solution contains points in one quadrant of the coordinate plane.
The solution contains points in two quadrants of the coordinate plane.
The solution contains points in three quadrants of the coordinate plane.
The solution contains points in all four quadrants of the coordinate plane.

Answer 1

Rewrite the system as
y ≤ 1.75(x − 5) = 1.75x − 8.75
y ≤ x^2 − 8.

So y must be ≤ the smaller of the two expressions.

- Quadrant I (x>0,y>0): take x = 6, y = 1. Then 1 ≤ 1.75(6−5)=1.75 and 1 ≤ 6^2−8=28, so (6,1) satisfies the system. So points in quadrant I occur.
- Quadrant II (x<0,y>0): impossible, because for x<0 the line 1.75x−8.75 < −8.75 < 0, so the minimum of the two bounds is <0 and no positive y can satisfy y ≤ (negative number).
- Quadrant III (x<0,y<0): take x = −1, y = −11. Then −11 ≤ 1.75(−1−5)=−10.5 and −11 ≤ (−1)^2−8=−7, so (−1,−11) works.
- Quadrant IV (x>0,y<0): take x = 1, y = −9. Then −9 ≤ 1.75(1−5)=−7 and −9 ≤ 1−8=−7, so (1,−9) works.

Thus the solution set contains points in three quadrants (I, III and IV).