## If $$p, s,$$ and $$t$$ are positive integer, is $$|ps - pt| > p(s - t)?$$

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### If $$p, s,$$ and $$t$$ are positive integer, is $$|ps - pt| > p(s - t)?$$

by VJesus12 » Wed Aug 18, 2021 7:30 am

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## Global Stats

If $$p, s,$$ and $$t$$ are positive integer, is $$|ps - pt| > p(s - t)?$$

(1) $$p < s$$
(2) $$s < t$$

Source: Official Guide

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### Re: If $$p, s,$$ and $$t$$ are positive integer, is $$|ps - pt| > p(s - t)?$$

by [email protected] » Thu Aug 19, 2021 7:48 am

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## Global Stats

VJesus12 wrote:
Wed Aug 18, 2021 7:30 am
If $$p, s,$$ and $$t$$ are positive integer, is $$|ps - pt| > p(s - t)?$$

(1) $$p < s$$
(2) $$s < t$$

Source: Official Guide
Target question: Is |ps - pt| > p(s - t) ?
In other words, Is |ps - pt| > ps - pt?
This is a good candidate for rephrasing the target question.

KEY CONCEPT: |x - y| can be thought as the DISTANCE between x and y on the number line.
For example, |3 - 10| = the DISTANCE between 3 and 10 on the number line.
And |6 - 1| = the DISTANCE between 6 and 1 on the number line.

IMPORTANT: We can also find the distance 6 and 1 on the number line by simply subtracting 6 - 1 to get 5, so why do we need absolute values? Can't we just conclude that |x - y| = x - y?
Great questions, me!
For SOME values of x and y, it's true that |x - y| = x - y, and for other values it is NOT the case that |x - y| = x - y
For example, if x = 5 and y = 2, then we get: |5 - 2| = 5 - 2. In this case |x - y| = x - y
Likewise, if x = 11 and y = 3, then we get: |11 - 3| = 11 - 3. In this case |x - y| = x - y
And, if x = 7 and y = 7, then we get: |7 - 7| = 7 - 7. In this case |x - y| = x - y

CONVERSELY, if x = 4 and y = 6, then we get: |4 - 6| = 4 - 6. In this case |x - y| x - y
Likewise, if x = 5 and y = 20, then we get: |5 - 20| = 5 - 20. In this case |x - y| x - y
And, if x = 0 and y = 1, then we get: |0 - 1| = 0 - 1. In this case |x - y| x - y

Notice the |x - y| = x - y IS true when x > y, and |x - y| = x - y is NOT true when x < y

If x < y, then |x - y| = some POSITIVE value, and x - y = some NEGATIVE value.
This means that, if x < y, then |x - y| > x - y
The target question asks Is |ps - pt| > ps - pt?

According to our conclusion above, if ps > pt, then |ps - pt| = ps - pt and . . .
if ps < pt, then |ps - pt| > ps - pt

This means we can REPHRASE the target question....
REPHRASED target question: Is ps < pt?

We can make things even easier, if we notice that, since p is POSITIVE, we can safely take the inequality ps < pt and divide both sides by p to get: s < t?.

RE-REPHRASED target question: Is s < t?

At this point, it will be very easy to analyze the answer choices....

Statement 1: p < s
Since there's no information about t, there's no way to answer the RE-REPHRASED target question with certainty.
Statement 1 is SUFFICIENT

Statement 2: s < t
Perfect!
The answer to the RE-REPHRASED target question is YES, s IS less than t
Since we can answer the RE-REPHRASED target question with certainty, statement 2 is SUFFICIENT