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Re: Are the integers z and f to the right of 0 on the number
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21 Mar 2012, 12:11
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Expert Reply
Are the integers z and f to the right of 0 on the number line?
Question basically asks whether z and f are both positive.
(1) The product of z and f is positive --> zf>0 --> z and f have the same sign, they are either both positive or both negative. Not sufficient.
(2) The sum of z and f is positive --> z+f>0 --> it's certainly possible that both are positive but it's also possible one to be positive and another negative (for example z=2 and f=-1). Not sufficient.
(1)+(2) Since from (2) z+f>0 then from (1) it's not possible that z and f to be both negative. Hence, z and f are both positive. Sufficient.
Are the integers z and f to the right of 0 on the number
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Updated on: 17 May 2021, 08:45
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Expert Reply
Top Contributor
enigma123 wrote:
Are the integers z and f to the right of 0 on the number line?
(1) The product of z and f is positive. (2) The sum of z and f is positive.
Target question:Are the integers z and f to the right of 0 on the number line? This is a good candidate for rephrasing the target question.
If z and f to the right of 0 on the number line, then that means z and f are both POSITIVE Let's now REPHRASE the target question.... REPHRASED target question:Are integers z and f both POSITIVE?
Aside: the video below has tips on rephrasing the target question
Statement 1: The product of z and f is positive. There are two possible scenarios that satisfy statement 1: Case a: z and f are both positive. In this case, the answer to the REPHRASED target question is YES, z and f are both POSITIVE Case b: z and f are both negative. In this case, the answer to the REPHRASED target question is NO, z and f are not both POSITIVE Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: The sum of z and f is positive. There are infinitely-many values of z and f that satisfy statement 2. Here are two: Case a: z = 1 and f = 1. In this case, the answer to the REPHRASED target question is YES, z and f are both POSITIVE Case b: z = -1 and f = 2. In this case, the answer to the REPHRASED target question is NO, z and f are not both POSITIVE Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined From statement 1, we concluded that EITHER z and f are both positive OR z and f are both negative If z and f are both positive, then the sum of z and f will be POSITIVE If z and f are both negative, then the sum of z and f will be NEGATIVE
Statement 2 tells us that The sum of z and f is positive. AHA! This means it must be the case that z and f are both positive So, the answer to the REPHRASED target question is YES, z and f are both POSITIVE Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT
Re: Are the integers z and f to the right of 0 on the number
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12 Oct 2019, 16:06
Expert Reply
enigma123 wrote:
Are the integers z and f to the right of 0 on the number line?
(1) The product of z and f is positive. (2) The sum of z and f is positive.
In other words, we need to determine: are both z and f positive?
Statement One Alone:
The product of z and f is positive.
Since the product of z and f is positive, either z and f are both positive or both negative. Thus, we cannot determine on which side of the number line they fall. Statement one is not sufficient to answer the question.
Statement Two Alone:
The sum of z and f is positive.
Although the sum is positive, either both z and f are positive or one is positive and one is negative. Thus, we cannot determine on which side of the number line they fall. Statement two is not sufficient to answer the question.
Statements One and Two Together:
Using the statements together, we see that the only possible scenario that works is when both z and f are positive.
Re: Are the integers z and f to the right of 0 on the number
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26 Nov 2020, 15:57
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