Question: In the xy- plane, region R consists of all the points (x, y) such that \(2x+3y\leq6\). Is the point (r, s) in the region R?

1. 3r + 2s = 6
2. \(r\leq3\) and \(s\leq2\)

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

Answer:
Solution with Explanation:
Approach Solution (1):

(1) 3r + 2s = 6 --- very easy to see that this statement is not sufficient:

If r = 2 and s = 0 then 2r + 3s = 4 < 6, so the answer is YES;
If r = 0 and s = 3 then 2r + 3s = 9 > 6, so the answer is NO.
Not sufficient

(2) \(r\leq3\) and \(s\leq2\) --- also very easy to see that this statement is not sufficient:

If r = 0 and s = 0 then 2r + 3s = 0 < 6, so the answer is YES;
If r = 3 and s = 2 then 2r + 3s = 12 > 6, so the answer is NO.
Not sufficient
(1) + (2) we already have an example for YES answer in (1) which is valid for the combined statements:
If r = 2 < 3 and s = 0 < 2 then 2r + 3s = 4 < 6, so the answer is YES;

To get NO answer try maximum possible value of s, which is s = 2, then from (1) r =\(\frac{2}{3}\) < 3 --- 2r + 3s = \(\frac{4}{3}\)+ 6 > 6, so the answer is NO
Not sufficient

Correct Option: E

Approach Solution (2):

Given: Region R consists of all the points (x, y) such that \(2x+3y\leq6\)
So, what does the region R look like?
To find out, let’s first graph the equation, 2x + 3y = 6

Since Region R is described as an inequality, we can choose any point on the coordinate plane to test whether or not it is in Region R. An easy point to test is (0, 0)
So, does x = 0 and y = 0 satisfy the inequality \(2x+3y\leq6\) ? YES
2(0) + 3(0) is less than or equal to 6.
So, the point (0, 0) is in the Region R. More importantly, every point on the same side of the line will also be in the Region R.

Statement 1: 3r + 2s = 6
The target question refers to the point (r, s)
In other words, the x- coordinate is r and the y- coordinate is s.
So, all the points (r, s) that satisfy the above equation can be found on the line 3x + 2y = 6
In other words, statement 1 tells us that the point (r, s) lies somewhere on the red line below.

As you can see, some points are in Region R, and some points are not in region R
Since we cannot answer the target question with certainty, statement 1 is not sufficient.
Statement 2: \(r\leq3\) and \(s\leq2\)
There are many points that satisfy this condition.
In fact, the point (r, s) can be anywhere inside the red box shown below.

As you can see, some points are in the Region R, and some points are not in Region R
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Statement 1 and Statement 2 combined
When we combine the statements, we are saying that the point (r, s) is on the red line (2x + 3y = 6) and inside the red box.

As you can see by the two blue points below, it’s possible to have a point in Region R, and it’s possible to have a point not in Region R

Since we cannot answer the target question with certainty, the combined statements are NOT Sufficient.

Correct Option: E

“In the xy- plane, region R consists of all the points (x, y) such that \(2x+3y\leq6\). Is the point (r, s) in the region R?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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