Picking Numbers

Some questions on the ACT can be made easier by plugging in real numbers in place of variables. Picking numbers can give a student a more intuitive understanding of the problem, instead of keeping things very theoretical.

When choosing numbers for a problem, it can be helpful to try and identify: what is the core mathematical concept the problem is trying to touch on?

For example, let’s look at Problem #41 from the December 2020 ACT.

This problem can be thought about conceptually (meaning leaving the variables in rather than testing actual numbers), but that can be hard to understand sometimes. To make it easier, a student can test out some values and see if they can figure out which statement is true for all values of x. Looking at the possible answers to the question, it is easy to notice that there are four different answers that involve the absolute value.

Since absolute value equations change negatives to positive, it is likely worthwhile to choose a negative number to test. For example, a student can pick the number -2. Once they’ve got the test number, they can start testing to see if each answer is true for the test value (if it’s false for the test value, then it is not true for all values like the question stipulates.)

It’s important to note that test values can prove that a statement is not true, but they cannot necessarily prove that a statement is true.

 − x < x  − (−2)  <  − 2 2  <  − 2

This statement is not true for all values.

 − x < |−x|  − (−2) < |−(−2)| 2 < |2| 2  < 2

This statement is not true for all values

x = |x|  − 2 = |−2|  − 2 = 2

This statement is not true for all values.

|x| = |−x| |−2| = |−(−2)| |−2| = |2| 2 = 2

This statement is true for at least our test value (not necessarily all values)

 − |x| = |−x|  − |−2| = |−(−2)|  − |−2| = |2|  − 2  = 2

This statement is not true for all values.

The test value shows that A, B, C, and E are not true for all values, which means that D must be the answer. Again, it’s worth noting that just because D worked at our test value, it didn’t necessarily mean it had to be true for all values. If a student had chosen positive two (2) instead of negative two (-2), some of the statements above would have been true. This is why it was important to choose a negative value, so it was clear how the absolute values affected the outcome.