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# Which of the following inequalities has a solution set, when

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Which of the following inequalities has a solution set, when [#permalink]  03 Jan 2011, 06:06
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Which of the following inequalities has a solution set, when graphed on the number line, is a single line segment of finite length?

A. x^4 >= 1
B. x^3 <= 27
C. x^2 >= 16
D. 2 <= |x| <= 5
E. 2 <= 3x+4 <= 6
[Reveal] Spoiler: OA

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Re: which of the following inequalities has a solution that [#permalink]  03 Jan 2011, 06:41
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anilnandyala wrote:
which of the following inequalities has a solution that , when graphed on the number line is a straight line of finite length?

Which of the following inequalities has a solution set, when graphed on the number line, is a single line segment of finite length?

The key words in the stem are: "a single line segment of finite length"

Now, answer choices A, B, and C can not be correct answers as solutions sets for these exponential functions are not limited at all (>= for even powers and <= for odd power) and thus can not be finite (x can go to + or -infinity for A and C and x can got to -infinity for B). As for D: we have that absolute value of x is between two positive values, thus the solution set for x (because of absolute value) will be two line segments which will be mirror images of each other.

Just to demonstrate:

A. x^4 >= 1 --> $$x\leq{-1}$$ or $$x\geq{1}$$: two infinite ranges;

B. x^3 <= 27 --> $$x\leq{3}$$: one infinite range;

C. x^2 >= 16 --> $$x\leq{-4}$$ or $$x\geq{4}$$: two infinite ranges;

D. 2 <= |x| <= 5 --> $$-5\leq{x}\leq{-2}$$ or $$2\leq{x}\leq{5}$$: two finite ranges;

E. 2 <= 3x+4 <= 6 --> $$-2\leq{3x}\leq{2}$$ --> $$-\frac{2}{3}\leq{x}\leq{\frac{2}{3}}$$: one finite range.

Hope it's clear.
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Re: which of the following inequalities has a solution that [#permalink]  11 Jun 2011, 21:36
It is good question,
It was obvious that A,B , C are incorrect as these are exponent of X but I couldn't figure out which one between d & e is better, so attempted D on GMAT Prep test
Later, During review of the question I found that X was actually |X| - absolute value , Hence two lines,
So correct is E
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Re: which of the following inequalities has a solution that [#permalink]  11 Jun 2011, 21:59
A. x^4 ≥ 1 --> or : two infinite ranges;

B. x^3 ≤ 27 --> : one infinite range;

C. x^2 ≥ 16--> or : two infinite ranges;

D. 2 ≤ |x| ≤ 5--> or : two finite ranges;

E. 2 ≤ 3x+4 ≤ 6 -->2-4 ≤ 3x+4-4 ≤ 6-4
--> -2 ≤ 3x ≤ 2 --> -2/3 ≤x ≤ 2/3 : one finite range.
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Re: Which of the following inequalities has a solution set, when [#permalink]  21 Oct 2013, 08:03
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Re: Which of the following inequalities has a solution set, when [#permalink]  13 Jul 2014, 07:34
Bunuel wrote:
anilnandyala wrote:
which of the following inequalities has a solution that , when graphed on the number line is a straight line of finite length?

Which of the following inequalities has a solution set, when graphed on the number line, is a single line segment of finite length?

The key words in the stem are: "a single line segment of finite length"

Now, answer choices A, B, and C can not be correct answers as solutions sets for these exponential functions are not limited at all (>= for even powers and <= for odd power) and thus can not be finite (x can go to + or -infinity for A and C and x can got to -infinity for B). As for D: we have that absolute value of x is between two positive values, thus the solution set for x (because of absolute value) will be two line segments which will be mirror images of each other.

Just to demonstrate:

A. x^4 >= 1 --> $$x\leq{-1}$$ or $$x\geq{1}$$: two infinite ranges;

B. x^3 <= 27 --> $$x\leq{3}$$: one infinite range;
C. x^2 >= 16 --> $$x\leq{-4}$$ or $$x\geq{4}$$: two infinite ranges;

D. 2 <= |x| <= 5 --> $$-5\leq{x}\leq{-2}$$ or $$2\leq{x}\leq{5}$$: two finite ranges;

E. 2 <= 3x+4 <= 6 --> $$-2\leq{3x}\leq{2}$$ --> $$-\frac{2}{3}\leq{x}\leq{\frac{2}{3}}$$: one finite range.

Hope it's clear.

Hi Bunuel, for option B, why isn't it a finite range?

x^3<=27
-3<=x<=3
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Re: Which of the following inequalities has a solution set, when [#permalink]  13 Jul 2014, 07:36
Expert's post
russ9 wrote:
Bunuel wrote:
anilnandyala wrote:
which of the following inequalities has a solution that , when graphed on the number line is a straight line of finite length?

Which of the following inequalities has a solution set, when graphed on the number line, is a single line segment of finite length?

The key words in the stem are: "a single line segment of finite length"

Now, answer choices A, B, and C can not be correct answers as solutions sets for these exponential functions are not limited at all (>= for even powers and <= for odd power) and thus can not be finite (x can go to + or -infinity for A and C and x can got to -infinity for B). As for D: we have that absolute value of x is between two positive values, thus the solution set for x (because of absolute value) will be two line segments which will be mirror images of each other.

Just to demonstrate:

A. x^4 >= 1 --> $$x\leq{-1}$$ or $$x\geq{1}$$: two infinite ranges;

B. x^3 <= 27 --> $$x\leq{3}$$: one infinite range;
C. x^2 >= 16 --> $$x\leq{-4}$$ or $$x\geq{4}$$: two infinite ranges;

D. 2 <= |x| <= 5 --> $$-5\leq{x}\leq{-2}$$ or $$2\leq{x}\leq{5}$$: two finite ranges;

E. 2 <= 3x+4 <= 6 --> $$-2\leq{3x}\leq{2}$$ --> $$-\frac{2}{3}\leq{x}\leq{\frac{2}{3}}$$: one finite range.

Hope it's clear.

Hi Bunuel, for option B, why isn't it a finite range?

x^3<=27
-3<=x<=3

x^3<=27 --> $$x\leq{3}$$.
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Which of the following inequalities has a solution set, when [#permalink]  18 Feb 2015, 04:35
Bunuel wrote:
anilnandyala wrote:
which of the following inequalities has a solution that , when graphed on the number line is a straight line of finite length?

Which of the following inequalities has a solution set, when graphed on the number line, is a single line segment of finite length?

The key words in the stem are: "a single line segment of finite length"

Now, answer choices A, B, and C can not be correct answers as solutions sets for these exponential functions are not limited at all (>= for even powers and <= for odd power) and thus can not be finite (x can go to + or -infinity for A and C and x can got to -infinity for B). As for D: we have that absolute value of x is between two positive values, thus the solution set for x (because of absolute value) will be two line segments which will be mirror images of each other.

Just to demonstrate:

A. x^4 >= 1 --> $$x\leq{-1}$$ or $$x\geq{1}$$: two infinite ranges;

B. x^3 <= 27 --> $$x\leq{3}$$: one infinite range;

C. x^2 >= 16 --> $$x\leq{-4}$$ or $$x\geq{4}$$: two infinite ranges;

D. 2 <= |x| <= 5 --> $$-5\leq{x}\leq{-2}$$ or $$2\leq{x}\leq{5}$$: two finite ranges;

E. 2 <= 3x+4 <= 6 --> $$-2\leq{3x}\leq{2}$$ --> $$-\frac{2}{3}\leq{x}\leq{\frac{2}{3}}$$: one finite range.

Hope it's clear.

Could someone explain Option A in detail?
I understand upon taking 4th root on both sides it becomes: x>= +-1
But, I don't understand how it gets simplified further as its been explained as: x^4 >= 1 --> $$x\leq{-1}$$ or $$x\geq{1}$$
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Re: Which of the following inequalities has a solution set, when [#permalink]  18 Feb 2015, 04:40
Expert's post
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connectvinoth wrote:
Bunuel wrote:
anilnandyala wrote:
which of the following inequalities has a solution that , when graphed on the number line is a straight line of finite length?

Which of the following inequalities has a solution set, when graphed on the number line, is a single line segment of finite length?

The key words in the stem are: "a single line segment of finite length"

Now, answer choices A, B, and C can not be correct answers as solutions sets for these exponential functions are not limited at all (>= for even powers and <= for odd power) and thus can not be finite (x can go to + or -infinity for A and C and x can got to -infinity for B). As for D: we have that absolute value of x is between two positive values, thus the solution set for x (because of absolute value) will be two line segments which will be mirror images of each other.

Just to demonstrate:

A. x^4 >= 1 --> $$x\leq{-1}$$ or $$x\geq{1}$$: two infinite ranges;

B. x^3 <= 27 --> $$x\leq{3}$$: one infinite range;

C. x^2 >= 16 --> $$x\leq{-4}$$ or $$x\geq{4}$$: two infinite ranges;

D. 2 <= |x| <= 5 --> $$-5\leq{x}\leq{-2}$$ or $$2\leq{x}\leq{5}$$: two finite ranges;

E. 2 <= 3x+4 <= 6 --> $$-2\leq{3x}\leq{2}$$ --> $$-\frac{2}{3}\leq{x}\leq{\frac{2}{3}}$$: one finite range.

Hope it's clear.

Could someone explain Option A in detail?
I understand upon taking 4th root on both sides it becomes: x>= +-1
But, I don't understand how it gets simplified further as its been explained as: x^4 >= 1 --> $$x\leq{-1}$$ or $$x\geq{1}$$

x >= +/- 1 does not make any sense.

When taking 4th root from both sides we'll get $$|x| \geq{1}$$, which is the same as $$x\leq{-1}$$ or $$x\geq{1}$$.

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Re: Which of the following inequalities has a solution set, when [#permalink]  18 Apr 2015, 08:52
Bunuel , Could you please elaborate on how to solve
2 <= |x| <= 5 --> −5≤x≤−2 or 2≤x≤5

in steps..?
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Re: Which of the following inequalities has a solution set, when [#permalink]  19 Apr 2015, 02:53
Expert's post
Ralphcuisak wrote:
Bunuel , Could you please elaborate on how to solve
2 <= |x| <= 5 --> −5≤x≤−2 or 2≤x≤5

in steps..?

2 <= |x| <= 5:

1. 2 <= x <= 5

2. 2 <= -x <= 5 --> multiply by -1: -2 >= x >= -5
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Re: Which of the following inequalities has a solution set, when   [#permalink] 19 Apr 2015, 02:53
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