For this problem, the limitation given to us is that  must be less than or equal to . Therefore we first attempt to plug in  into both Quantity A and B to determine which is larger.

Quantity A:

Quantity B:   

Quantity B is larger.

Plugging in 

Quantity A: 

Quantity B:   

Quantity B is still larger.

Therefore we can conclude for whenever  is positive, Quantity B will be larger.

We notice that because Quantity A is raised to the power of 3 and Quantity B is raised to the power of 1, any time we plug in a negative number, Quantity A and B will be negative.  However, Quantity A will always be more neagtive because it is raised to the power of 3.  Therefore for all negative numbers, Quantity B will be larger.

Therefore, whenever , Quantity B will be larger than Quantity A

Work from the "inside" outward.  Therefore, first solve 4 # 1 by replacing a with 4 and b with 1:

4 # 1 = (4 * 1) + 4 = 4 + 4 = 8

That means: 3 # (4 # 1) = 3 # 8.  Solve this now:

3 # 8 = (3 * 8) + 3 = 24 + 3 = 27

Step 1: 7x + 3y

Step 2: 4 * (7x + 3y) = 28x + 12y

Step 3: 28x + 12y + x = 29x + 12y

Step 4: 29x + 12y – (x – y) = 29x + 12y – x + y = 28x + 13y

Quantity A: (x – y)² = x² – 2xy + y²

Quantity B: (x + y)² = x² + 2xy + y²

Both have x² + y² so cancel those from both columns and just compare –2xy in Quantity A to 2xy in Quantity B. If x = 1 and y = 1, –2xy = –2 and 2xy = 2, so Quantity B is greater. But if x = 1 and y = –1, –2xy = 2 and 2xy = –2, so Quantity A is greater. The contradiction means the answer cannot be determined.

If the first integer is , then 

This is the same as the median.

The expression is simplified as follows:

Since  the value of  must be slightly greater for it to be 17 when raised to the 4th power.

(x + y)² = x² + 2xy + y²

Now, since there are no specifications on what x and y can equal, one or both of them could be 0, making the two columns equal. Any value other than 0 will make the columns unequal because of the additional 2xy term, so the answer cannot be determined.

Use substitution to determine the relationship.

For example, we could plug in  and .

So far it looks like the first expression is greater, but it's a good idea to try other values of x and y to be sure. This time, we'll try some negative values, say,  and .

This time the first quantity is smaller. Therefore the relationship cannot be determined from the information given.

We have three variables and only two equations, so we will not be able to solve for each independent variable. We need to think of another solution.

Notice what happens if we line up the two equations and add them together. 

(x + y) + (3x – y + z) = 4x + z

and 5 + 3 = 8

Lets take this equation and multiply the whole thing by 3:

3(4x + z = 8)

Thus, 12x + 3z = 24.

Plugging in numbers is not the best strategy here.  Instead, let's see how we can equate the two expressions.  Quantity A is actually a difference of squares.  256x² = (16x)² and 49y² = (7y)².  These look like the expressions in Quantity B.  The formula to remember here is the difference of squares formula, a very important one for this test!  a² – b² = (a + b)(a – b). Thus, if a = 16x and b = 7y, 256x² – 49y² = (16x – 7y)(16x + 7y), and the quantities are equal.