Math: 34847f8a

ID: 34847f8a 

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)

Comment: Welcome back to Algebra I.

Method 1: To add fractions, find a common denominator. The right-hand side of the given equation provides the solution: (x - 2)(x + 5). To ensure that the left-hand side will be in the same terms, multiply each numerator by the missing part in the denominator:

2/(x - 2) * (x + 5)/(x + 5) + 3/(x + 5) * (x - 2)/(x - 2) = (rx + t)/(x - 2)(x + 5)

We can now ignore the denominator, knowing that everything will be the same, and proceed with the algebra in the numerator to eventually solve for rt.

2(x + 5) + 3(x - 2) = rx + t

2x + 10 + 3x - 6 = rx + t

5x + 4 = rx + t

r = 5 and t = 4

Hence, rt = (5)(4), or 20. The correct answer is (C).

Method 2: You can use Desmos to model two functions, setting each side of the given equation equal to "y." Adding sliders for "r" and "t" can prove overwhelming, though, which is why I would suggest the algebraic method above. If you are up for an exploration, though, jump in and see what you can discover.

As you drag the slider for "r" to the right, the blue curve overlaps with the bottom-left red curve around 4.5; it aligns with the upper-right red curve around 5.8. This is not the most encouraging sign, since there are an infinite number of values in between as well as another slider to manipulate, but in terms of integers, only 5 fits in between. It makes sense to set the slider to 5.

Now, play with the slider for "t" as well. You will find that the two graphs overlap when t = 4.

The two functions are now the same, and if r = 5 and t = 4, then rt = 20, and the correct answer is (C).

Method 3: You could choose values for "x" and run through some calculations, but with two more unknowns and only the hint that both "r" and "t" are positive constants—not integers—it would not be the best use of time. Still, you could use the answers to test different integer combinations of "r" and "t," since they are all integer answers. An educated guess is better than a complete guess.

A. -20 makes no sense, since the product of two positive constants cannot be negative. Get rid of this option immediately.

B. 15 has positive integer factors in 1 and 15 or 3 and 5, nothing else.

C. 20 has positive integer factors in 1 and 20, 2 and 10, and 4 and 5.

D. 60 has positive integer factors in 1 and 60, 2 and 30, 3 and 20, 4 and 15, 5 and 12, and 6 and 10. Phew! That is a lot of potential work.

Since the setup tells you that the given equation is true for all x > 2, you can plug in something like 3 to test. Then, you can isolate "r" and "t." I would strongly suggest leaving (D) alone and working with simpler combinations of factors.

2/((3) - 2) + 3/((3) + 5) = (r(3) + t)/((3) - 2)((3) + 5)

2/1 + 3/8 = (3r + t)/(1)(8)

16/8 + 3/8 = (3r + t)/8

19 = 3r + t

Looking at answer choice (B), neither combination will work:

19 ≠ 3(1) + (15), which is 18; 3(15) + (1), which is 46; 3(3) + (5), which is 14; or 3(5) + (3), which is 18

You can deduce that (B) is not likely the answer. Run through the combinations from (C) above in the same manner.

19 ≠ 3(1) + (20), which is 23; 3(20) + (1), which is 61; 3(2) + (10), which is 16; or 3(10) + (2), which is 32

However, notice what happens with 4 and 5:

19 ≠ 3(4) + (5), which is 17, but 19 = 3(5) + (4)

(C) is more promising. To avoid testing so many possibilities for (D), we can observe a truth from the above: it must be true that for 3r + t to equal 19, if "r" and "t" are integers, then one must be odd and the other even. We know this because an odd plus an odd is even, and of course, an even plus an even is even. 3 * odd is odd, while 3 * even is even. We can at least cut out the two combinations of even numbers, 2 and 30 and 6 and 10. There is also no way for 1 and 60 to fit into the equation, since either way, 3r + t would be greater than 19; the same could be said of 3 and 20, 4 and 15, or 6 and 10. The least value that could come of any of these combinations is 27: 3(4) + (15) or 3(5) + (12). In any case, (D) is in the same camp as (B) was, with no integer values that work. It would be best to pick the safest option in (C) and move on.

Please note, as all the digital ink above would suggest, I would not recommend solving the question in this manner. Method 1 above is the most viable option. This and method 2 above were thought exercises, good to play around with, not so good on a timed exam.

Error Alert: There is an error in the official explanation (OE) of this question that, as of this writing, has not been corrected. The OE recommends "multiplying the first expression by 

x + 5/x - 5," rather than (x + 5)/(x + 5), to create a common denominator.