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

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

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02 Mar 2009, 20:47
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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

OPEN DISCUSSION OF THIS QUESTION IS HERE: which-of-the-following-inequalities-has-a-solution-set-that-127820.html
[Reveal] Spoiler: OA
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Re: Which of the following inequalities has a solution set, when graphed [#permalink]

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03 Mar 2009, 03:34
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E

1. Any expressions that contains only x in the form of |x|, x^2, x^4, x^2n are insensitive to sign of x (A,C,D in our case). Therefore, zero must satisfy such expressions, otherwise we will have at least one hole near zero and two segments. So, check x=0 for all three options. None of them fits requirement. So, A,C,D are out and B, E remain.

2. in B x=-inf satisfy the expression, so it doesn't represent finite segments.

3. Only E remains. 3x + 4 is a line cut in points x=2, and x=6 --> a finite segment.
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Kudos [?]: 512 [0], given: 99

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

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03 Mar 2009, 09:02
walker,
Amazing explanation. I did follow the same way what you explained about A, C, D but I chose B and didn't realize it can satisfy infinite also ...

now it is clear to me.

(+1) kudos to you walker.

Thank you.

Last edited by ugimba on 03 Mar 2009, 09:11, edited 1 time in total.
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Re: Which of the following inequalities has a solution set, when graphed [#permalink]

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03 Mar 2009, 09:06
and one more question, I made 9 mistakes in quant when I write gmatprep and still end up making 50. When I retook the exam and made just 3 mistakes only and still made 50. why it happend? it is huge range for 50 then ( from 9 mistakes to 3 mistakes in my observation)? so to get 51, there should be no wrongs at all? have to make 37 out 37 corrects..?
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Kudos [?]: 3061 [0], given: 360

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

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03 Mar 2009, 17:19
ugimba wrote:
and one more question, I made 9 mistakes in quant when I write gmatprep and still end up making 50. When I retook the exam and made just 3 mistakes only and still made 50. why it happend? it is huge range for 50 then ( from 9 mistakes to 3 mistakes in my observation)? so to get 51, there should be no wrongs at all? have to make 37 out 37 corrects..?

A few mistakes (I had 4 mistakes in my prep and 51) at the end of the test still give you chance to get 51.
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Re: Which of the following inequalities has a solution set, when graphed [#permalink]

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04 Mar 2009, 12:51
walker wrote:
E

1. Any expressions that contains only x in the form of |x|, x^2, x^4, x^2n are insensitive to sign of x (A,C,D in our case). Therefore, zero must satisfy such expressions, otherwise we will have at least one hole near zero and two segments. So, check x=0 for all three options. None of them fits requirement. So, A,C,D are out and B, E remain.

2. in B x=-inf satisfy the expression, so it doesn't represent finite segments.

3. Only E remains. 3x + 4 is a line cut in points x=2, and x=6 --> a finite segment.

Hi Walker, could you please explain two things (sorry if they are too naive):

How do you check if "inf" satisfies an expression?

How did you figure out the cut points for 3x+4 ?

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

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04 Mar 2009, 13:23
krishan wrote:
How do you check if "inf" satisfies an expression?

There is a nice concept: when x=inf (or x=-inf) there is no need to calculate complex expression.
For example, y=-8x^8 + x^6 +30 x^3 +4x +2000
at x=inf (or a very huge number) we choose only the biggest power and omit all constants.
So, our complex expression becomes a simple one: y = -x^8 and at x=-inf, y=-inf.
And again, think about inf as a huge number, let's say 1000000000000000

krishan wrote:
How did you figure out the cut points for 3x+4 ?

y=3x+4 is a line. Just draw any line and cut it by two y=a and y=b lines (a,b - any numbers), you will get a segment.
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Re: Which of the following inequalities has a solution set, when graphed [#permalink]

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04 Mar 2009, 14:00
thanks a lot Walker..
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Re: Which of the following inequalities has a solution set, when graphed [#permalink]

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02 Nov 2010, 06:38
which of the following inequalities have a solution set that , when a graphed on the number line is a single line segment of finate length

a x^4 >= 1
b x^3 <= 27
c x^2 >= 16
d 2 <= mod(x) <= 15
e 2 <= 3x+4 <= 6
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Re: Which of the following inequalities has a solution set, when graphed [#permalink]

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02 Nov 2010, 06:45
anilnandyala wrote:
which of the following inequalities have a solution set that , when a graphed on the number line is a single line segment of finate length

a x^4 >= 1
b x^3 <= 27
c x^2 >= 16
d 2 <= mod(x) <= 15
e 2 <= 3x+4 <= 6

E. You can easily eliminate the other four options:

A) This is true for any value of x such that $$x \leq -1$$ or $$x \geq 1$$ - two line segments of infinite length.
B) This is true for all $$x \leq 3$$ - infinite length.
C) Like (A), this is true for all $$x \leq -4$$ or $$x \geq 4$$.
D) True for $$-15 \leq x \leq -2$$ or $$2 \leq x \leq 15$$.
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Re: Which of the following inequalities has a solution set, when graphed [#permalink]

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19 Mar 2016, 14:40
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Re: Which of the following inequalities has a solution set, when graphed [#permalink]

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20 Mar 2016, 02:12
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ugimba 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

OPEN DISCUSSION OF THIS QUESTION IS HERE: which-of-the-following-inequalities-has-a-solution-set-that-127820.html

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.

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

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20 Mar 2016, 02:14
Bunuel wrote:
ugimba 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

OPEN DISCUSSION OF THIS QUESTION IS HERE: which-of-the-following-inequalities-has-a-solution-set-that-127820.html

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.

OPEN DISCUSSION OF THIS QUESTION IS HERE: which-of-the-following-inequalities-has-a-solution-set-that-127820.html

SIMILAR QUESTION TO PRACTICE: which-of-the-following-inequalities-has-a-solution-set-that-130666.html

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Re: Which of the following inequalities has a solution set, when graphed   [#permalink] 20 Mar 2016, 02:14
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