How many integer ordered (x,y) pairs represent possible solutions to a set of (x,y) condition inequalities?

Question

How many ordered pairs of integers represent possible solutions to the set of conditions:

y > (1/10)x + (1/10)

y < 5 – 0.99x

y < -1 + 1.1x

Answer 1

You may either plot the inequalities and determine that a domain and range from 0 to 5 is sufficient or you may use trial-and-error values, -100 to 100.

FOR x = -100 TO 100

FOR y = -100 TO 100

IF ( y > (x/10 + 1/10) AND ( y < (5-0.99*x) ) AND ( y < (-1+1.1*x) ) ) THEN OUTPUT x,y

NEXT y

NEXT x

2 1

3 1

3 2

4 1

Don't be a;alarmed -- this program ran for one second.

Someday soon, computers (even on smart phones) will be like calculators and exhaustive enumeration (considering every possibility) will be an acceptable mathematics solution !

Answer 2

To get an initial idea of the solution, you can graph this. All three inequalities are in slope-intercept form, and you can check your graph with Desmos or a graphing calculator. Once you make the graph, you will see that there is a small triangular region where all three inequalities are true. The following ordered pairs of integers exist within this region: (2,1), (3,1), (4,1), (3,2). Two of these, (4,1) and (3,2) are at the edge of the solution region, so you might not see them with an initial visual inspection!

You can check that each solution is valid by plugging it back into the set of inequalities.

For (3,2):

2 > (1/10)3+(1/10) → 2 > 2/5 → TRUE

2 < 5 – 0.99(3) → 2 < 2.03 → TRUE (but very close!)

2 < -1 + 1.1(3) → 2 < 2.3 → TRUE

The check for the other solutions are similar.

Answer 3

If we graph all three of these inequalities, we will find that the shaded region of points that make all three of these inequalities true is a small triangular region with vertices at about (1.1,0.2), (2.9,2.2), and (4.5,0.6).

The only integer pairs that fall in that region are:

(2,1), (3,1), (4,1), (3,2)

So there are four integer ordered pairs that are possible solutions to the inequalities. That means that if you plug in those ordered pairs into all of the inequalities, they will be true.

Answer 4

You can't go about this question piecemeal, that is, considering each inequality in isolation, since each inequality merely separates the (x,y) plane into 2 infinite sections!

Instead, express the inequalities as equations in standard form, graph them, and count the grid-points inside their intersection area. If you have a calculator that can graph multiple equations, fine; but, remember that you have to be the visual assigner of the inequality areas. Also, you should be proficient enough to sketch such a graph probably more rapidly than displaying on a calculator, anyway, and in a standardized testing condition, both speed and accuracy count.

You will obtain a triangular area containing obviously the gridpoints: (2,1), (3,1), (4,1), and (3,2). That's 4 ordered pairs.