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# If a and b are integers, is a < 0?

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Math Expert
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If a and b are integers, is a < 0?  [#permalink]

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28 Apr 2016, 03:48
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If a and b are integers, is a < 0?

(1) b^2 – a^2 < 0
(2) b^2 – a^3 > 0

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If a and b are integers, is a < 0?  [#permalink]

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Updated on: 29 Apr 2016, 07:13
The question ask us to check where a is a negative integer.

Stmnt 1:
b^2 – a^2 < 0
b^2<a^2
e.g 4 < a^2 so a can be both positive or negative
Insufficient

Stmnt 2:
b^2 – a^3 > 0
b^2 > a^3
e.g when a= 1 b=4 the answer is no
e.g when a=1 b=-1 the answer is yes
Insufficient

Stmnt 1+2
Adding the inequalities we get
b^2-a^2 < b^2-a^3
Editing my answer .I did not change sign when i multiplied it by -1
a^2 > a^3
a^3 >a^2
This states that 'a' can never be negative.The answer to the question is NO
So a can only be negative.However the answer remains same

Sufficient

Ans C

Originally posted by raarun on 28 Apr 2016, 04:58.
Last edited by raarun on 29 Apr 2016, 07:13, edited 2 times in total.
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Re: If a and b are integers, is a < 0?  [#permalink]

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28 Apr 2016, 09:57
I will go for E

wse cant just add equalities without knowig sign of variables
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Re: If a and b are integers, is a < 0?  [#permalink]

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28 Apr 2016, 11:53
Condition 1: b^2 -a^2 < 0
simplifying this equation gives us (b-a)*(b+a)<0.
We can form different cases where a and b assume -ve and +ve values and find that equation is possible under multiple circumstances.
Thus this data is insufficient.

Condition 2: b^2-a^3 > 0

we cant form 2 different cases where both a and b are +ve and -ve and 2 cases where one of them is -ve and find that second condition holds for multiple cases. Thus insufficient.

Combining both the conditions, we find that a^3 is less than a^2. This is only possible when either a <0 or 0<a<1. since a and b both are integers, as given in the question. a<0 is the only possibility.

Hence C is answer.
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If a and b are integers, is a < 0?  [#permalink]

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28 Apr 2016, 13:27
Bunuel wrote:
If a and b are integers, is a < 0?

(1) b^2 – a^2 < 0
(2) b^2 – a^3 > 0

1) This just means that $$a^2 > b^2.$$ We still don't know anything about specific values for a or b. For $$a = 2, b = 1$$ we have FALSE, but for $$a = -2, b = 1$$ we have TRUE. Insufficient.

2) Similar to above, this just means that $$a^3 < b^2.$$ This doesn't tell us anything about specific values of a or b. For $$a = 1, b = 2$$ we have FALSE, but for $$a = -1, b = 2$$ we have TRUE. Insufficient.

Together, we have that $$a^3 < b^2 < a^2$$, or that $$a^3 < a^2$$. Since a and b are both integers, the only possibility is that $$a<0.$$

Answer: C
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If a & b are integers, is a<0  [#permalink]

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26 Sep 2018, 06:17
If a & b are integers, is a<0
1. b^2 - a^2 < 0
2. b^2 - a^3 >0
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Re: If a & b are integers, is a<0  [#permalink]

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26 Sep 2018, 07:58
In stat 1 since b^2 - a^2 < 0 means a>b but we cannot say anything about its sign. Not sufficient
In stat 2 again we cannot infer anything about a's sign. For ex- b=4, a=2 satisfies it and b=4 and a=-2 also satisfies it
Taking the two stat together we can say that since a>b and b^2 - a^3 >0 and since a and b are integers a should be less than 0.
C should be the correct choice.
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If a & b are integers, is a<0  [#permalink]

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26 Sep 2018, 07:59
Abhi077 wrote:
If a & b are integers, is a<0
1. b^2 - a^2 < 0
2. b^2 - a^3 >0

1. b^2 - a^2 < 0

$$b^2 < a^2$$
b and a can be both positive and negative

not sufficient

2. b^2 - a^3 >0

$$b^2 > a^3$$

b = 1 and a =1/2 => a>0
b=10 and a=-50 => a<0

not sufficient

using both

$$b^2 > a^3$$
$$b^2 < a^2$$
is only possible when a is negative

sufficient
C
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Joined: 02 Sep 2009
Posts: 51167
Re: If a and b are integers, is a < 0?  [#permalink]

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26 Sep 2018, 11:09
Abhi077 wrote:
If a & b are integers, is a<0
1. b^2 - a^2 < 0
2. b^2 - a^3 >0

Merging topics.

Please search before posting and name topics properly. Check rules here: https://gmatclub.com/forum/rules-for-po ... 33935.html Thank you.
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Re: If a and b are integers, is a < 0?  [#permalink]

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26 Sep 2018, 12:23
Bunuel wrote:
If a and b are integers, is a < 0?

(1) b^2 – a^2 < 0
(2) b^2 – a^3 > 0

$$a,b\,\,{\text{ints}}$$

$${\text{a}}\,\,\mathop {\text{ < }}\limits^? \,\,{\text{0}}$$

$$\left( 1 \right)\,\,\,{a^2} > {b^2}\,\,\,\left[ {\, \geqslant 0\,\,\,\, \Rightarrow \,\,\,{a^2} > 0\,\,\, \Rightarrow \,\,\,a \ne 0} \right]\,\,\,\,\,\,\left( * \right)$$

$$\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {a,b} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\ \,{\text{Take}}\,\,\left( {a,b} \right) = \left( { - 1,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\ \end{gathered} \right.$$

$$\left( 2 \right)\,\,{b^2} > {a^3}\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {a,b} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\ \,{\text{Take}}\,\,\left( {a,b} \right) = \left( { - 1,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\ \end{gathered} \right.$$

$$\left( {1 + 2} \right)\,\,\,{a^2} > {b^2} > {a^3}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \begin{gathered} {a^2} > {a^3} \hfill \\ \left( * \right)\,\,a \ne 0 \hfill \\ \end{gathered} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,{a^2}\,\, > \,\,0} \,\,\,1 > a\,\,\,\,\,\mathop \Rightarrow \limits^{a\,\,\operatorname{int} \, \ne \,\,0} \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If a and b are integers, is a < 0? &nbs [#permalink] 26 Sep 2018, 12:23
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# If a and b are integers, is a < 0?

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