There is a particular ACT math question that always raises students’ eyebrows because of an unusual answer choice that they can’t decipher.

The question goes:

Which of the following is the set of all real numbers x such that x+3 > x+5?

(F)The empty set

(G)The set containing all real numbers

(H)The set containing all negative real numbers

(J)The set containing all nonnegative real numbers

(K)The set containing only zero

Alright, so as you probably guessed, students often don’t know what answer (F) means.  First of all, let’s try to solve this problem and figure out the answer without thinking about the answer choices.


x+3 > x+5 (x on both sides, so maybe subtract x to get it by itself)

-x         -x

3 > 5

Well that’s weird.  Usually when we solve equations in math we end up with “x equals something” or “x is greater or less than something.”  When we end up with something like 3>5 (or 5 = 6, or anything else that’s NOT true) the answer is no solutions.  Definitely a good thing to know for the SAT and ACT (by the way, if you end up with something like 6 > 2 or 5=5 or something that IS true, the answer is infinite solutions or all real numbers).

Now, many of my students are familiar with this rule, and therefore know that this inequality has no solutions.  If you get this far, you should absolutely be able to get the right answer here, even if you have no idea what answer (F) means.

If I look at the other answer choices, I am 100% sure that none of those things is the same thing as “no solution.”  If you’re not sure, you could still test numbers in each set to see if they work. For example, if I wanted to test answer (K), I could just see if zero works as an answer. (0+3 is NOT greater than 0+5, so zero is NOT an answer here).

This type of situation often arises with vocab questions.  Students know that 3 answers are wrong, and aren’t sure what the fourth word means.  If that ever happens, PICK THE WORD YOU DON’T KNOW!  Students often feel nervous doing that, but if you know the other answers are wrong, you should be confident picking the one you don’t understand.

Similarly here, even if you don’t know what “the empty set” means (by the way, it’s just another way of saying “no solutions” because the solution set is empty), you should definitely pick that if you know the other answers are wrong.

This strategy of picking an answer you don’t know if you’re sure that the other answers are wrong is definitely helpful, so don’t be afraid to use it!

Till next time!


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