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# Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2

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Math Expert
Joined: 02 Sep 2009
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Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2  [#permalink]

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13 Mar 2018, 06:56
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Difficulty:

55% (hard)

Question Stats:

51% (01:28) correct 49% (01:31) wrong based on 53 sessions

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Is z + z < z ?

(1) -4z > 4z
(2) z^3 < z^2

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Math Expert
Joined: 02 Sep 2009
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Re: Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2  [#permalink]

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13 Mar 2018, 06:57
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Re: Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2  [#permalink]

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13 Mar 2018, 07:09
IMO A.

2z < z

If we can certainly tell if z if either positive or negative we can answer the question.

St 1 : -4z > 4z
This is only true for z < 0.
Hence suff.

St 2 : z^3 < z^2

Two possibilities :

1) z could be negative
Hence z^3 ( neg no.) < z^2 ( positive no.)

2) z could be positive and less than 1.

Z^3 < z^2 still holds.

Not sufficient .

Hence A.

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Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2  [#permalink]

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13 Mar 2018, 10:10
1
Bunuel wrote:
Is z + z < z ?

(1) -4z > 4z
(2) z^3 < z^2

Question: Is $$z + z < z ?$$

or Is $$2z - z < 0 ?$$

or Is $$z < 0 ?$$

St 1:$$- 4z > 4z$$

or $$8z < 0$$

or $$z < 0$$ Sufficient

St 2 : $$z^3 < z^2$$

or $$z^3 - z^2 < 0$$

or $$z^2 ( z - 1) < 0$$

$$z^2 > 0$$ for $$z < 0$$ or $$z > 0$$

$$z - 1 < 0$$

$$z < 1$$

Therefore combined range is $$z < 0$$ or $$0 < z < 1.$$ Not sufficient

(A)
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Re: Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2  [#permalink]

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13 Mar 2018, 10:29
1. -4z>4z => z is +ve hence 2z>z ---->S
2. z^3 <z^2 => z is -ve or 0<z<1
if z is -ve => 2z<z ---->S
but if z is -ve or 0<z<1 =>2z>z ---->NS Thus 2. is NS

Hence I would go for option A.
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Re: Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2  [#permalink]

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15 Mar 2018, 13:03
Bunuel wrote:
Is z + z < z ?

(1) -4z > 4z
(2) z^3 < z^2

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

The question is modified as follows.
2z < z ?
⇔ z < 0 ?

Condition 1)
-4z > 4z
⇔ 0 > 8z
⇔ 0 > z
It is same as the question.
Condition 1) is sufficient.

Condition 2)
z^3 < z^2
⇔ z^3 - z^2 < 0
⇔ z^2(z-1) < 0
⇔ z - 1 < 0 since z^2 > 0
⇔ z < 1
Since the range of the question does not include that of the condition 2), the condition 2) is not sufficient.

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Re: Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2  [#permalink]

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15 Mar 2018, 13:10
Bunuel wrote:
Is z + z < z ?

(1) -4z > 4z
(2) z^3 < z^2

Rephrase the question:

Is z <0??

(1) -4z > 4z

8z < 0....Z must be negative...........Answer is always Yes.

Sufficient

(2) z^3 < z^2

Let x = 1/2 ............1/8 < 1/4 ..........Answer is NO
Let x =-1/2 .......... -1/8 < 1/4 ..........Answer is Yes

Insufficient

Re: Is z + z < z ? (1) -4z > 4z (2) z^3 < z^2   [#permalink] 15 Mar 2018, 13:10
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