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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]

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03 Jan 2011, 07:06
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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]

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03 Jan 2011, 07:41
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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]

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11 Jun 2011, 22:36
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]

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11 Jun 2011, 22:59
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]

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13 Jul 2014, 08:34
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]

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13 Jul 2014, 08:36
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]

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18 Feb 2015, 05:35
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]

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18 Feb 2015, 05:40
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]

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18 Apr 2015, 09:52
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]

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19 Apr 2015, 03:53
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]

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05 Sep 2016, 18:23
Ralphcuisak wrote:
Bunuel , Could you please elaborate on how to solve
2 <= |x| <= 5 --> −5≤x≤−2 or 2≤x≤5

in steps..?

The absolute value of X is between 2 and 5.
X = 2 through 5
X = -2 through -5

On the number line in bold:

--|---|--0--|---|--
---2--5--0--2---5--

2 separate finite line segments.

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Re: Which of the following inequalities has a solution set, when [#permalink]

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22 Apr 2017, 05:14
What makes the answer E line segment a finite length?

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Re: Which of the following inequalities has a solution set, when [#permalink]

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09 Aug 2017, 11:32
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,

I think I am unable to understand the question here. I was between B and E and chose B because it gave one value which is 3.

I failed to understand what "a single line segment of finite length" means. Can you explain this and what we actually need to find here?

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Re: Which of the following inequalities has a solution set, when [#permalink]

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09 Aug 2017, 23:31
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Expert's post
pra1785 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,

I think I am unable to understand the question here. I was between B and E and chose B because it gave one value which is 3.

I failed to understand what "a single line segment of finite length" means. Can you explain this and what we actually need to find here?

$$x^3 \leq 27$$ gives $$x\leq{3}$$, not x = 3.

Maybe images could help:

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.

Similar questions:
https://gmatclub.com/forum/which-of-the ... 27820.html (GMAT Prep).
https://gmatclub.com/forum/which-of-the ... 30666.html (Knewton).

[Reveal] Spoiler:
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Re: Which of the following inequalities has a solution set, when [#permalink]

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anilnandyala wrote:
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

We must determine from our answer choices which inequality is a single line segment of finite length when graphed on the number line.

A) x^4 ≥ 1

∜(x^4) ≥ ∜1

|x| ≥ 1

x ≥ 1 or x ≤ -1

Since the solution set from answer choice A produces two rays of infinite length, answer choice A is not correct.

B) x^3 ≤ 27

∛(x^3) ≤ ∛27

x ≤ 3

Since the solution set from answer choice B produces one ray of infinite length, answer choice B is not correct.

C) x^4 ≥ 16

∜(x^4) ≥ ∜16

|x| ≥ 2

x ≥ 2 or x ≤ -2

Since the solution set from answer choice C produces two rays of infinite length, answer choice C is not correct.

D) 2 ≤ |x| ≤ 5

We must solve for when x is positive and for when x is negative.

When x is positive:

2 ≤ x ≤ 5

When x is negative:

2 ≤ -x ≤ 5

-1(2 ≤ -x ≤ 5)

-2 ≥ x ≥ -5

Since the solution set from answer choice D produces two line segments of finite length, answer choice D is not correct.

Since answers A, B, C, and D are not correct, E must be the correct answer. However, for practice, we can solve it anyway.

E) 2 ≤ 3x + 4 ≤ 6

-2 ≤ 3x ≤ 2

-2/3 ≤ x ≤ 2/3

Since the solution set from answer choice E produces one line segment of finite length, answer choice E is correct.

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Re: Which of the following inequalities has a solution set, when [#permalink]

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15 Aug 2017, 13:34
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Hi,
Since question is saying single line of finite length,
Graph is going to be a straight line is equation is a linear equation. Linear equation is equation with highest power of variable being one. For higher powers, graph is going to be a curve.
Option A has highest power of x as 4, so it is curve
Option B has highest power of x as 3, so it is also curve
Option C has highest power of x as 2, so it is also curve
Option D has absolute function. An absolute function is combination of two functions, so it is going to give two lines.
Thus option E
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Re: Which of the following inequalities has a solution set, when   [#permalink] 15 Aug 2017, 13:34
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