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

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Bunuel
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If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru [#permalink] New post   25 May 2020, 05:18
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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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GMATinsight
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If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru [#permalink] New post   Updated on: 29 Jun 2020, 19:49
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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



i.e. Answer: Option E

Originally posted by GMATinsight on 25 May 2020, 05:34.
Last edited by GMATinsight on 29 Jun 2020, 19:49, 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] New post   Updated on: 25 May 2020, 06:32
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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.

Originally posted by GMATWhizTeam on 25 May 2020, 05:48.
Last edited by GMATWhizTeam on 25 May 2020, 06:32, edited 1 time in total.
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Re: If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru [#permalink] New post   25 May 2020, 20:22
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See the attachment.
Answer E
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Bunuel
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Re: If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru [#permalink] New post   25 May 2020, 06:00
Expert Reply
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.

____________________
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] New post   25 May 2020, 06:01
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] New post   25 May 2020, 09:53
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



i.e. Answer: Option E



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] New post   27 Jun 2022, 05:32
is there another way to solve this:2x^2–3x+1>0 ?
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Re: If 0.7^(2x2 - 3x + 4) < 0.343, then which of the following must be tru [#permalink]
27 Jun 2022, 05:32
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