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

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

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03 Jan 2011, 07:06
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Which of the following inequalities has a solution set that, when graphed on the number line, is a single line segment of finite length?

A. $$x^4 \geq 1$$

B. $$x^3 \leq 27$$

C. $$x^2\geq 16$$

D. $$2 \leq |x| \leq 5$$

E. $$2 \leq 3x+4 \leq 6$$

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

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19 Feb 2012, 21:24
12
11
dvinoth86 wrote:
Which of the following inequalities has a solution set that when graphed on the number line, is a single segment of finite length?

A. x4 ≥ 1
B. x3 ≤ 27
C. x2 ≥ 16
D. 2≤ |x| ≤ 5
E. 2 ≤ 3x+4 ≤ 6

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.

Answer: E.

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.

Answer: E.

Hope it's clear.
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Re: Which of the following inequalities has a solution set that  [#permalink]

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30 Jan 2013, 03:58
1
3
mridulparashar1 wrote:
Which of the following inequalities has a solution set that when graphed on the number line, is a single
segment of finite length?
A. $$x^4 \geq 1$$
B. $$x^3 \leq 27$$
C. $$x^2 \geq 16$$
D. $$2\leq |x| \leq 5$$
E. $$2 \leq 3x+4 \leq 6$$

Question taken from one of the Quant files in the download section at Gmatclub.

In all the above options, we are going to get graphs with range values. Does this Questions asks where the range is limited/minimum (Finite Length)

How would you solve Option D.

Thanks
Mridul

The question asks for the option for which the range does not extend to infinity or does not have a break in between.

A) x can be any value greater than 1
B) x can be any value lesser than 3
C) x can be any value greater than 4
D) This option does have a finite range. However, there is a break inbetween for values in the range -2 < x < 2.
E) Same as $$-\frac{2}{3} \leq x \leq \frac{2}{3}$$. Finite straight line.
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Re: Which of the following inequalities has a solution set that  [#permalink]

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10 Jul 2013, 10:57
2
Which of the following inequalities has a solution set that when graphed on the number line, is a single segment of finite length?

A. x^4 ≥ 1
4√x^4 ≥ 4√1
x ≥ 1
x is found anywhere from 1 to + infinity. INVALID

B. x^3 ≤ 27
3√x^3 ≤ 3√27
x ≤ 3
x is found anywhere from 3 to -infinity. INVALID

C. x^2 ≥ 16
√x^2 ≥ √16
|x| ≥ 4
x≥4 OR x≤-4
x is found anywhere from 4 to +infinity or from -4 to -infinity. INVALID

D. 2≤ |x| ≤ 5
2≤|x|
2≤x OR -2≥x
|x| ≤ 5
x≤5 OR x≥-5
This option has two segments of finite length, not one as is required by the question. INVALID

E. 2 ≤ 3x+4 ≤ 6
-2 ≤ 3x ≤ 2
(-2/3) ≤ x ≤ (2/3)
x is found anywhere from (-2/3) to (2/3). VALID

(E)
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Which of the following inequalities has a solution set that  [#permalink]

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23 Jun 2014, 01:20
1
1
Similar question to practice: http://gmatclub.com/forum/which-of-the- ... 30666.html
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Re: Which of the following inequalities has a solution set that, 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.

Answer: E.

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.

Answer: E.

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 that, when  [#permalink]

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18 Feb 2015, 05:40
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.

Answer: E.

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.

Answer: E.

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 that, 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 that, 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 that  [#permalink]

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20 Apr 2015, 01:13
1
Bunuel wrote:
dvinoth86 wrote:
Which of the following inequalities has a solution set that when graphed on the number line, is a single segment of finite length?

A. x4 ≥ 1
B. x3 ≤ 27
C. x2 ≥ 16
D. 2≤ |x| ≤ 5
E. 2 ≤ 3x+4 ≤ 6

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.

Answer: E.

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.

Answer: E.

Hope it's clear.

Dear Bunuel,

Sir , if option B is a one infinite range....then even this ans can be correct like E .
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Re: Which of the following inequalities has a solution set that  [#permalink]

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20 Apr 2015, 04:32
Capt wrote:
Bunuel wrote:
dvinoth86 wrote:
Which of the following inequalities has a solution set that when graphed on the number line, is a single segment of finite length?

A. x4 ≥ 1
B. x3 ≤ 27
C. x2 ≥ 16
D. 2≤ |x| ≤ 5
E. 2 ≤ 3x+4 ≤ 6

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.

Answer: E.

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.

Answer: E.

Hope it's clear.

Dear Bunuel,

Sir , if option B is a one infinite range....then even this ans can be correct like E .

How is $$-\frac{2}{3}\leq{x}\leq{\frac{2}{3}}$$ an infinite range?
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Which of the following inequalities has a solution set that  [#permalink]

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25 Jul 2016, 04:36
1
Which of the following inequalities has a solution set that when graphed on the number line, is a single line segment of finite length?

$$A. x^4 ≥ 1$$

$$B. x^3 ≤ 27$$

$$C. x^2 ≥ 16$$

$$D. 2≤ |x| ≤ 5$$

$$E. 2 ≤ 3x+4 ≤ 6$$

Lets check one by one
$$x^4 ≥ 1$$

Nope :- This maps to a two value of x≥1 and x ≤-1 on the number line. This is the not a finite line. This is actually an infinite line broken only from -1 to 1 but extending from -∞ to +∞

$$x^3 ≤ 27$$

Nope:- This maps to infinite values of ≤3 on the number line. This is the equation of a INFINITE line from 3 to -∞

$$x^2 ≥ 16$$

Nope :- This maps to a two value of x≥4 and x ≤-4 on the number line. This is the not a finite line. This is actually an infinite line broken only from -4 to 4 but otherwise extending from -∞ to +∞

$$2≤ |x| ≤ 5$$

Looks interesting. Lets check |X| can actually be seen as +x and also =-x
Lets check both cases individually

CASE 1) $$2≤ x≤ 5$$ This gives a finite line.

CASE 2) $$2≤ -x ≤ 5$$ or$$-5≤ x ≤ -2$$This gives a finite line again
A TOTAL OF 2 FINITE RANGES. We need only one finite range.

$$2 ≤ 3x+4 ≤ 6$$

$$x≥-\frac{2}{3}$$ and $$x≤\frac{2}{3}$$ or $$-\frac{2}{3}≤x≤\frac{2}{3}$$

YES this is a finite range from -x to +x

E IS THE ANSWER

≥ ≤ ∞

dvinoth86 wrote:
Which of the following inequalities has a solution set that 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

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Re: Which of the following inequalities has a solution set that, 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.

Answer: E.

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.

Answer: E.

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

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09 Aug 2017, 23:31
3
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.

Answer: E.

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.

Answer: E.

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).

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

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15 Aug 2017, 11:00
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.

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

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15 Aug 2017, 13:34
2
1
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 that, when  [#permalink]

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06 Feb 2018, 15:35
Top Contributor
1
anilnandyala wrote:
Which of the following inequalities has a solution set that, when graphed on the number line, is a single line segment of finite length?

A. $$x^4 \geq 1$$

B. $$x^3 \leq 27$$

C. $$x^2\geq 16$$

D. $$2 \leq |x| \leq 5$$

E. $$2 \leq 3x+4 \leq 6$$

Attachment:
Untitled.pdf

IMPORTANT
This is one of those questions that require us to check/test the answer choices. In these situations, always check the answer choices from E to A, because the correct answer is typically closer to the bottom than to the top.

E) 2 ≤ 3x + 4 ≤ 6
Subtract 4 from all sides to get: -2 ≤ 3x ≤ 2
Divide all sides by 3 to get: -2/3 ≤ x ≤ 2/3
So, x can have any value from -2/3 to 2/3
So, if we were to graph the possible values of x, the line segment would have a FINITE length.

Answer: E

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

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17 Mar 2018, 12:43
Hi All,

The equations in this question can be physically drawn or handled as a Number Property concept:

Note that the question asks for a SINGLE SEGMENT of FINITE LENGTH. This means that the group of numbers that make up the solution set would be consecutive and limited (as opposed to non-consecutive and/or infinite).

Here are the Number Properties:

A: X^4 >= 1

X can be 1, 2, 3, etc. or -1, -2, -3, etc. This group of answers is infinite and non-consecutive. Eliminate A.

B: X^3 <= 27

X can be 3, 2, 1, 0, -1, etc. This group is infinite. Eliminate B.

C: X^2 >= 16

X can be 4, 5, 6, etc. or -4, -5, -6, etc. This group is infinite and non-consecutive. Eliminate C.

D: 2 <= |X| <= 5

X can be 2, 3, 4, 5 or -2, -3, -4, -5, This group is finite BUT non-consecutive. Eliminate D.

E: 2 <= 3X + 4 <= 6

Let's simplify with algebra…

-2 <= 3X <= 2

-2/3 <= X <= 2/3

X is a finite set of answers and the answers are consecutive. This is a MATCH to what we're looking for.

Final Answer:

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

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Re: Which of the following inequalities has a solution set that,   [#permalink] 07 Oct 2018, 00:33
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