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# If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru

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Joined: 02 Sep 2009
Posts: 65187
If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru  [#permalink]

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25 May 2020, 04:18
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If $$0.7^{(2x^2 - 3x + 4)} < 0.343$$, then which of the following must be true?

A. x < -1/2
B. -1/2 < x < 1/2
C. 1/2 < x < 1
D. x > 1
E. x < 1/2 or x > 1

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If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru  [#permalink]

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Updated on: 29 Jun 2020, 18:49
2
1
Bunuel wrote:
If $$0.7^{(2x^2 - 3x + 4)} < 0.343$$, then which of the following must be true?

A. x < -1/2
B. -1/2 < x < 1/2
C. 1/2 < x < 1
D. x > 1
E. x < 1/2

RULE:
If $$0 < a < 1$$, then $$a^x < a^y$$ if $$x > y$$

i.e. $$0.7^{(2x^2 - 3x + 4)} < 0.343$$
i.e. $$0.7^{(2x^2 - 3x + 4)} < (0.7)^3$$

i.e. $$(2x^2 - 3x + 4) > 3$$
i.e. $$(2x^2 - 3x + 1) > 0$$

i.e. $$(2x^2 - 2x - x + 1) > 0$$

i.e. $$(2x - 1)*(x - 1) > 0$$

i.e. either both (2x - 1)*(x - 1) should be positive or both should be Negative

For both positive, x>1

For both negative, x<1/2

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Originally posted by GMATinsight on 25 May 2020, 04:34.
Last edited by GMATinsight on 29 Jun 2020, 18:49, edited 2 times in total.
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If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru  [#permalink]

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Updated on: 25 May 2020, 05:32
1
Bunuel wrote:
If $$0.7^{(2x^2 - 3x + 4)} < 0.343$$, then which of the following must be true?

A. x < -1/2
B. -1/2 < x < 1/2
C. 1/2 < x < 1
D. x > 1
E. x < 1/2

Solution

• $$0.7^{(2x^2 – 3x + 4)} < 0.342$$
$$⟹0.7^{(2x^2 – 3x + 4)} < 0.7^3$$
• Since, 0.7 < 1 , so higher power of 0.7 will result in lower value.
o Therefore, $$2x^2 – 3x + 4 > 3$$
$$⟹ 2x^2 – 3x + 1 > 0$$
$$⟹ 2x^2 – 2x -x + 1> 0$$
$$⟹ 2x(x – 1) – 1(x – 1)> 0$$
$$⟹ (x – 1)(2x – 1) > 0$$

o Thus, $$x < \frac{1}{2}$$ or $$x > 1$$

Thus, the correct answer is Option E.
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Originally posted by GMATWhizTeam on 25 May 2020, 04:48.
Last edited by GMATWhizTeam on 25 May 2020, 05:32, edited 1 time in total.
Math Expert
Joined: 02 Sep 2009
Posts: 65187
Re: If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru  [#permalink]

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25 May 2020, 05:00
GMATinsight wrote:
Bunuel wrote:
If $$0.7^{(2x^2 - 3x + 4)} < 0.343$$, then which of the following must be true?

A. x < -1/2
B. -1/2 < x < 1/2
C. 1/2 < x < 1
D. x > 1
E. x < 1/2

RULE:
If $$0 < a < 1$$, then $$a^x < a^y$$ if $$y > x$$

i.e. $$0.7^{(2x^2 - 3x + 4)} < 0.343$$
i.e. $$0.7^{(2x^2 - 3x + 4)} < (0.7)^3$$

i.e. $$(2x^2 - 3x + 4) > 3$$
i.e. $$(2x^2 - 3x + 1) > 0$$

i.e. $$(2x^2 - 2x - x + 1) > 0$$

i.e. $$(2x - 1)*(x - 1) > 0$$

i.e. either both (2x - 1)*(x - 1) should be positive or both should be Negative

For both positive, x>1

For both negative, x<1/2

No match of answer... Let me check again

Bunuel I think the inequation sign should be reversed in this question. Please check.

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Edited option E. Thank you.
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Re: If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru  [#permalink]

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25 May 2020, 05:01
D and/or E

0.343 = 0.7^3

Also we know that :
If 0<a<1 , then for x>y a^x < a^y

==> 2x^2 -3x +4 >3
or 2x^2 - 3x +1 >0
or (2x-1)(x-1)> 0

==> X< 1/2 (Option E) or x>1 (Option D)

Am I wrong somewhere?
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If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru  [#permalink]

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25 May 2020, 08:53
GMATinsight wrote:
Bunuel wrote:
If $$0.7^{(2x^2 - 3x + 4)} < 0.343$$, then which of the following must be true?

A. x < -1/2
B. -1/2 < x < 1/2
C. 1/2 < x < 1
D. x > 1
E. x < 1/2

RULE:
If $$0 < a < 1$$, then $$a^x < a^y$$ if $$y > x$$

i.e. $$0.7^{(2x^2 - 3x + 4)} < 0.343$$
i.e. $$0.7^{(2x^2 - 3x + 4)} < (0.7)^3$$

i.e. $$(2x^2 - 3x + 4) > 3$$
i.e. $$(2x^2 - 3x + 1) > 0$$

i.e. $$(2x^2 - 2x - x + 1) > 0$$

i.e. $$(2x - 1)*(x - 1) > 0$$

i.e. either both (2x - 1)*(x - 1) should be positive or both should be Negative

For both positive, x>1

For both negative, x<1/2

GMATinsight Please correct my understanding --> If $$0 < a < 1$$, then $$a^x < a^y$$ if $$y > x$$

In the above rule, Shouldn't be y>x as x>y ???
Also guide me how did the sign (<) of the original equation $$0.7^{(2x^2 - 3x + 4)} < 0.343$$ changed to (>) in the new equation $$(2x^2 - 3x + 4) > 3$$.
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Re: If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru  [#permalink]

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25 May 2020, 19:22
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Re: If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru   [#permalink] 25 May 2020, 19:22