SAT Math Multiple Choice Practice Question 435: Answer and Explanation
18. If (a - 5)(b + 5) < 0, then which of the following must be true?
A. (a < 5) and (b<5) OR (a>5) and (b>5)
B. (a < 5) and (b>5) OR (a<5) and (b<5)
C. (a<5) and (b>5) OR (a>5) and (b<5)
D. (a < 5) and (b=5) OR (a=5) and (b>5)
E. (a > 5) and (b<5) OR (a>5) and (b>5)
Correct Answer: C
C If you have an equation with two unknowns, you must have TWO ... not one ... conditions simultaneously satisfied in order to make an absolute deterministic statement that is always true or always false:
So, we are given that (a-5)(b+5) < 0 we must use LOGIC to find an answer with IF/THEN, EITHER/OR, and/or BOTH/AND conditions to determine when the relation is ALWAYS TRUE.
As an example, look at the case when (a-5)(b+5)=0 then EITHER a = 5 OR b = -5 makes the equation TRUE.
However, if we require (a-5)(b+5) < 0 then EITHER [ BOTH (a-5) < 0 AND (b+5) > 0 ] .... OR .... [ BOTH (a-5) > 0 AND (b+5) < 0 ] must be TRUE
This is equivalent to saying IF (a-5)(b+5)< 0 THEN EITHER we have a (Negative #) x (Positive #) OR (Positive #) x (Negative #)
So, IF (a-5)(b+5) < 0 THEN EITHER [BOTH a<5 AND b>-5 must be TRUE] ... OR ... [ BOTH a>5 AND b<-5 MUST be TRUE ]
Given the correct requirement, you can see conditions in three of the answers that are only part of the "must" requirement. Because none of the answers do not have two EITHER/OR with BOTH/AND simultaneous conditions, they are ALL wrong. The "plug in" answers approach given in the "explanations", leaving answer C by (untested) default, is wrong.