If wz < 2, is z < 1?

If wz < 2, is z < 1?

(1) w > 2. Subtract \(w > 2\) from \(wz < 2\) (we can do it as the signs are in opposite direction):

\(wz-w<2-2\) --> \(w(z-1)<0\), as w is positive (given w>2), then the product to be negative \(z-1\) must be negative --> \(z-1<0\) --> \(z<1\). Sufficient.

(2) z < 2. If \(z=1.5>1\) and \(w=0\) (wz<2), then the answer is YES but if \(z=0<1\) and \(w=0\) (wz<0), then the answer is NO. Not sufficient.

Answer: A.
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Thanks Bunuel

Even i got A. However, the OA given in the GMAC paper test is D.

I guess the OA (like many others in the paper tests) is wrong

If wz < 2, is z < 1?

(1) w > 2
(2) z < 2

Sol.
opt.1 => w>2
i.e. z(something>2)<2
=> z<2/(something >2)
in the step above, R.H.S will always be <1. Hence Z<1

opt.2 => z<2
from this we can't say whether z will be less then 1.
Hence, A.

Here the simple way to conclude that statement 1 is sufficient is because w is always positive and greater than 2
for w=2 which it cant be => z=1
so as Z increases z will decrease => Z<1
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Target question: Is z < 1?

Given: wz < 2

Statement 1: w > 2
This means that w is POSITIVE, which means we can divide both sides of the given inequality, wz < 2, by w.
We get: z < 2/w
First, since w is POSITIVE, we know that 2/w is POSITIVE, which means z is POSITIVE
Second, since w > 2, we know that 2/w will be less than 1, since the denominator is greater than the numerator
So, we can write 2/w < 1
Since z < 2/w, we can write: z < 2/w < 1
This means we can conclude that z < 1
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: z < 2
There are several values of w and z that satisfy statement 2 (and the given info). Here are two:
Case a: w = 1 and z = 0. Notice that this satisfies the given info that wz < 2. In this case z < 1
Case b: w = 1 and z = 1.5. Notice that this satisfies the given info that wz < 2. In this case z > 1
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer:

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It is given in (1) that w > 2

We are given that wz < 2

Since we know from (1) that w is positive, we can divide both sides of the inequality by w

=> z < 2/w

Now, since w > 2, this means that 2/w will be lesser than 1
=> z < 1

So, we can conclude that (1) is sufficient.

WZ < 2

Is Z < 1 ?

Statement 1: we can subtract the inequalities

WZ < 2
-(W > 2)
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WZ — W < 2 - 2

— Take common factor on the left hand side of inequality

W * (Z - 1) < 0

We know the Factors of: (W) and (Z - 1) must have OPPOSITE Signs for their product to be -Negative

Statement 1 tells us that W is greater than > +positive 2

Therefore, W must be = +positive value

Making the Factor (Z - 1) a -NEGATIVE Value

Z - 1 < 0

Z < 1

Statement 1 sufficient alone

S2:
Knowing that Z < 2 does not restrict the possible values of Z to only those less that’s < 1
Not sufficient

*A*

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