If b > 0 and ax < bx < by, is xy > 0?

Great question Brent. It is hard based on simple concepts and organization of data.

If b > 0 and ax < bx < by, is xy > 0?

(1) a – b > 0

a>b>0.........It means both a & b are Positive.

Hence, x must be Negative to yield that ax<bx

But we do not know anything bout sign of 'y'.

Let b=1 and x=-1 and y=10.........xy<0

Let b=1 and x=-1 and y=-1/2.........xy>0

Insufficient

(2) a < y

No clue about signs x, b, y and a can take many combinations with signs could verify that ax < by

Let a = 1 & y = 2 & x=b=1...... xy>0

Let x=-10 & y=1 & a=2 & b=-1...xy<0

Insufficient

Combining 1 & 2

y> a>b>0 & x<0....... then xy is NOT greater than Zero

Answer: C

Target question: Is xy > 0?

Given: b > 0 and ax < bx < by
Since we're told that a - b is POSITIVE, we can take bx < by, and divide both sides by b to get: x < y

Statement 1: a – b > 0
I can see (a - b) "hiding" in the inequality ax < bx, so let's look into this further.
Take ax < bx and subtract bx from both sides to get: ax - bx < 0
Factor: x(a - b) < 0
Since we're told that a - b is POSITIVE, we can divide both sides by a-b to get: x < 0
So, x is NEGATIVE
Also, a - b > 0, we can add b to both sides to get a > b
Is this enough information to answer the target question?
Well, we still don't know anything about y, so it seems unlikely that statement 1 is sufficient.
Let's TEST some numbers.
Here are two cases that both that satisfy statement 1:
Case a: a = 3, b = 2, x = -1 and y = 1. In this case, xy = -1. So, xy < 0
Case b: a = 3, b = 2, x = -1 and y = -0.5. In this case, xy = 0.5. So, xy > 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a < y
There are several scenarios that satisfy statement 2. Here are two:
Case a: a = 3, b = 10, x = 2 and y = 4. In this case, xy = 8. So, xy > 0
Case b: a = 2, b = 1, x = -1 and y = 10. In this case, xy = -10. So, xy < 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x is NEGATIVE and that b < a
Since it's given that 0 < b, we can write: x < 0 < b < a

Statement 2 tells us that a < y
If we add that information to the above inequality, we get: x < 0 < b < a < y
In other words, x is NEGATIVE and y is POSITIVE, which means xy < 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

RELATED VIDEO

Hi can you please tell me how x has to be negative since a and b are positive?

From stem we know ax<bx----->> x(a-b)<0---------(A)

from (1) we know a-b>0 thus for statement ----(A) to be valid , x must be <0

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