Testing Positive & Negative Cases

Some Positives & Negatives problems deal with multiple variables, each of which can be positive or negative. In these situations, you should set up a table listing all the possible positive/negative combinations of the variables, and determine what effect that would have on the question. For example:

If ab > 0, which of the following must be negative?

  1. a + b
  2. |a| + b
  3. b − a

One way to solve problems such as this one is to test numbers systematically. In this example, you can list both of the two possible positive/negative combinations of a and b that meet the criteria established in the question. Then, test each of the combinations in each of the answer choices. You can use a chart such as the one below to keep track of your work, choosing simple values (e.g., 3 and 6) to make calculations quickly:

Criterion:
ab > 0		 A
a + b
 B
| a | + b		 C
b − a		 D
 E
+ , +
a = 3
b = 6		  
YES		  
POS		  
POS		  
POS		  
POS		  
NEG
−,−
a = −3
b = −6		  
YES		  
NEG		  
NEG		  
NEG		  
POS		  
NEG

Notice that if more than one answer choice gives you the desired result for all cases, you can try another pair of numbers and test those answer choices again.

Another approach to this problem is to determine what you know from the fact that ab > 0. If ab > 0, then the signs of a and b must both be the same (both positive or both negative). This should lead you to answer choice (E), because  must be negative if a and b have the same sign.

Check Your Skills

  1. |x| > |y|. Which of the following must be true? Indicate all that apply.

    1. xy is positive
    2. x + y > 0
    3. x2 > y2

  2. If ab < 0, a > b, and a > −b, which of the following must be true?

    1. a/b > 0
    2. a + b < 0
    3. b − (−a) > 0
    4. a/b = 1
    5. a − b < 0