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If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes

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If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes  [#permalink]

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New post 21 Apr 2019, 05:40
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A
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C
D
E

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If \(\frac{x}{{(x – 2)(x + 1)}}\) ≤ 0, which of the following specifies all the possible values of x ?

A. x < –1

B. 0 ≤ x < 2

C. x ≤ –1 and 0 ≤ x < 2

D. x < –1 and 0 ≤ x < 2

E. x ≤ –1 and x ≥ 2

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Re: If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes  [#permalink]

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New post 21 Apr 2019, 06:14
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For such complex algebraic expressions, always look at the answer choices before attempting the question.

Anyway, for the expression \(\frac{x}{{(x – 2)(x + 1)}}\), x cannot be equal to -1 or 2. Therefore, eliminate answer choices C and E right away; these answer choices say that x ≤ –1, i.e. x = -1 is an accepted solution of x.

Now look closely at the three options. You will notice that option D is nothing but A + B combined. Therefore, quickly select two values of x that satisfy these expressions. x = -3 and x = 0 or 1/2 satisfy the expressions. Therefore, option D is the correct answer.

You should be able to complete this question within one minute if you approach strategically rather than solve the complex algebraic expression.
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If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes  [#permalink]

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New post Updated on: 22 Apr 2019, 19:21
DisciplinedPrep wrote:
For such complex algebraic expressions, always look at the answer choices before attempting the question.

Anyway, for the expression \(\frac{x}{{(x – 2)(x + 1)}}\), x cannot be equal to -1 or 2. Therefore, eliminate answer choices C and E right away; these answer choices say that x ≤ –1, i.e. x = -1 is an accepted solution of x.

Now look closely at the three options. You will notice that option D is nothing but A + B combined. Therefore, quickly select two values of x that satisfy these expressions. x = -3 and x = 0 or 1/2 satisfy the expressions. Therefore, option D is the correct answer.

You should be able to complete this question within one minute if you approach strategically rather than solve the complex algebraic expression.


That is an interesting approach. But considering that the given problem was already factored in such a great way to analyze, wouldn't it be just as fast to solve it graphically?

Originally posted by joaopschultz on 22 Apr 2019, 19:17.
Last edited by joaopschultz on 22 Apr 2019, 19:21, edited 2 times in total.
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If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes  [#permalink]

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New post 22 Apr 2019, 19:18
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Alternative approach:
Image
Answer: D
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Re: If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes  [#permalink]

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New post 22 Apr 2019, 20:30
joaopschultz wrote:
Alternative approach:
Image
Answer: D


Hey, should we not consider 2 cases here, where the first case would be when Numerator is greater than 0 and Denominator less than 0; the other would be vice versa.

Numerator and Denominator cannot both be positive or negative at the same time.
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If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes  [#permalink]

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New post 23 Apr 2019, 07:32
sumisachan wrote:
joaopschultz wrote:
Alternative approach:
Image
Answer: D


Hey, should we not consider 2 cases here, where the first case would be when Numerator is greater than 0 and Denominator less than 0; the other would be vice versa.

Numerator and Denominator cannot both be positive or negative at the same time.


Hey, sumisachan! We are, actually, considering all possible cases for each factor on the 4 relevant intervals:

\(( -∞, -1);\)
\([-1, 0);\)
\([0, 2);\)
\([2, ∞)\)

Our goal with this graphic analysis is to simplify the understanding of the whole function by breaking it into smaller problems which are how each factor of the function will behave for a given value of x(being positive or negative) during each of the relevant intervals.

With this evaluation our job becomes easier, since the function's behavior as a whole is a product of those factors. With that, the only job left is to analyze whether we include or not some critical points (such as -1 or 2, that are out of solution in order to make a denominator different from 0).

The real problem is not understanding the cases of positive/negative denominators or numerators, but rather the product of them.
That is why we evaluate the signals for each part and then multiply them to find the answer.
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Re: If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes  [#permalink]

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New post 25 May 2019, 04:39
I just plugged in the values provided in the options and checked whichever work. Also once you plug in you must remember to boundary condition. Like x = -1 and x=2 are 2 values which should be chosen as they lead to infinity.
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Re: If x/(x – 2)(x + 1) ≤ 0, which of the following specifoes   [#permalink] 25 May 2019, 04:39
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