For how many ordered pairs of positive integers (x,y) is 2x+3y<6?

a. One
b. Two
c. Three
d Five
e Seven

for 2x+3y=6, the x-intercept is 3, and the y-intercept is 2.

draw the line
you want the region below this line, bounded by the x and y axes.

How many whole number grid points do you see in that region?

Well, let me do some math while I juggle some clown noses... Oops, dropped one!

Now, let's solve the inequality 2x + 3y < 6 in a funny way.

If we imagine that x is 0, the inequality becomes 3y < 6, which gives us y < 2. So, we have one possible solution for the ordered pair (0,1).

Now, let's imagine that y is 0. We get 2x < 6, which simplifies to x < 3. This gives us three possible solutions for the ordered pairs (1,0), (2,0), and (3,0).

So, in total, we have four possible ordered pairs (0,1), (1,0), (2,0), and (3,0).

Oops, I almost forgot the option. So, the answer is d) Five! Oh, did I mention there was an extra clown nose in there? Silly me!

To find the number of ordered pairs of positive integers (x, y) that satisfy the inequality 2x + 3y < 6, we can use a systematic approach. The steps below outline the process:

Step 1: Start by substituting x = 1 in the inequality: 2(1) + 3y < 6.
Simplifying the inequality gives us 2 + 3y < 6.
Subtracting 2 from both sides, we get 3y < 4.

Step 2: Solve the inequality 3y < 4.
Dividing both sides by 3, we find y < 4/3.
Since y needs to be a positive integer, the largest possible value for y is 1.

Step 3: Repeat steps 1 and 2 for the values x = 2, 3, and so on until x = 6.

Using this approach, we can determine the number of possible values for y for each x:

For x = 1, y has only one possible value, y = 1.
For x = 2, y can take the values 1 and 2.
For x = 3, y can take the values 1 and 2.
For x = 4, y can take the values 1 and 2.
For x = 5, y can take the values 1 and 2.
For x = 6, y can take only one value, y = 1.

Thus, the total number of ordered pairs (x, y) that satisfy the given inequality is:

1 (from x = 1) + 2 (from x = 2 and 3) + 2 (from x = 4 and 5) + 1 (from x = 6) = 6.

Therefore, there are six ordered pairs of positive integers that satisfy the inequality 2x + 3y < 6. Hence, the answer is (c) Three.

The answer must be one. Thanks.

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