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Bunuel
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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 31 Jan 2021, 03:09
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95% (hard)

Question Stats:

35% (02:33) correct 65% (02:37) wrong based on 244 sessions

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If x is an integer less than or equal to 0 and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0
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A
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chetan2u
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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 01 Feb 2021, 01:04
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Bunuel
If x is an integer less than or equal to 0 and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0
Show more

As x is an integer less than or equal to 0, |x|=-x
\(|||x| - 3| - x| = 3\)
\(||-x - 3| - x| = 3\)

We will have two cases when we open the MODULUS |-x-3|: Either \(-x\geq{3}\) or \(-x<3\)

(I) If \(-x\geq 3\) or \(x\leq -3\), then -x-3>0, and
\(|-x - 3 - x| = 3\)
Square both sides => \(9+4x^2+12x=9..........4x(x+3)=0\)
So x=0 or x=-3
-3 is a valid solution.

(II) If -x<3 or x>-3, then -x-3<0, and
\(|3+x - x| = 3\)
So all values of x in this range will fit in => x={0, -1, -2}

Sum of all values = \((-3)+(-2)+(-1)+0=-6\)

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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 31 Jan 2021, 15:52
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Bunuel
If x is an integer and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0
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Bunuel There's a bug with this question, all integers greater than 3 can be the solution for this as the left-hand side would simplify to:

\(||x - 3| - x| = |x - 3 - x| = | -3| = 3\).
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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 31 Jan 2021, 19:47
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Bunuel
If x is an integer and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0
Show more

As also mentioned above, the solution for this would be \(x\geq{3}\) and \(0\geq{x}\geq{-3}\)

So sum will be (-3)+(-2)+(-1)+3+4+5....=Infinity
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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 01 Feb 2021, 00:31
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Bunuel
If x is an integer and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0

Bunuel There's a bug with this question, all integers greater than 3 can be the solution for this as the left-hand side would simplify to:

\(||x - 3| - x| = |x - 3 - x| = | -3| = 3\).
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Added the missing part to the question. Thank you.
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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 01 Feb 2021, 16:09
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Bunuel
If x is an integer less than or equal to 0 and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0

As x is an integer less than or equal to 0, |x|=-x
\(|||x| - 3| - x| = 3\)
\(||-x - 3| - x| = 3\)
(I) If -x>3 or x<-3, then -x-3>0, and
\(|-x - 3 - x| = 3\)
Square both sides => \(9+4x^2+12x=9..........4x(x+3)=0\)
So x=0 or x=-3
(II) If -x<3 or x>-3, then -x-3<0, and
\(|3+x - x| = 3\)
So all values of x in this range will fit in => x={0, -1, -2}

Sum of all values = (-3)+(-2)+(-1)+0=-6

A
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Hi, can you please explain as to how you made the two cases? I'm quite lost!

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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 01 Feb 2021, 19:06
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chetan2u
Bunuel
If x is an integer less than or equal to 0 and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0

As x is an integer less than or equal to 0, |x|=-x
\(|||x| - 3| - x| = 3\)
\(||-x - 3| - x| = 3\)
(I) If -x>3 or x<-3, then -x-3>0, and
\(|-x - 3 - x| = 3\)
Square both sides => \(9+4x^2+12x=9..........4x(x+3)=0\)
So x=0 or x=-3
(II) If -x<3 or x>-3, then -x-3<0, and
\(|3+x - x| = 3\)
So all values of x in this range will fit in => x={0, -1, -2}

Sum of all values = (-3)+(-2)+(-1)+0=-6

A


Hi, can you please explain as to how you made the two cases? I'm quite lost!

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As x is an integer less than or equal to 0, |x|=-x
\(|||x| - 3| - x| = 3\)
\(||-x - 3| - x| = 3\)

We will have two cases when we open the MODULUS |-x-3|: Either \(-x\geq{3}\) or \(-x<3\)
And the Modulus will behave differently in bot hthe cases.
If the -x-3<0, the mod when opened will have a '-' sign in front.
If -x-3>0, the mod will remain the same when opened.
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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 11 Feb 2021, 19:13
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(I) If -x>3 or x<-3, then -x-3>0, and
\(|-x - 3 - x| = 3\)
Square both sides => \(9+4x^2+12x=9..........4x(x+3)=0\)
So x=0 or x=-3
Show more

Neither 0 nor -3 are less than -3. Why isn't case 1 invalid?
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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 11 Feb 2021, 19:39
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chetan2u
If -x>3 or x<-3, then -x-3>0, and
\(|-x - 3 - x| = 3\)
Square both sides => \(9+4x^2+12x=9..........4x(x+3)=0\)
So x=0 or x=-3

Neither 0 nor -3 are less than -3. Why isn't case 1 invalid?
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Hi

We have to include -3 too as a value of the x in one of the two cases.
The first case was supposed to be -x is greater than or equal to 3.
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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 11 Feb 2021, 19:49
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If x is an integer and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0

As also mentioned above, the solution for this would be \(x\geq{3}\) and \(0\geq{x}\geq{-3}\)

So sum will be (-3)+(-2)+(-1)+3+4+5....=Infinity
Show more


It is mentioned in the question that X is integer less tha equal to zero. Hence possible values of X can only be -1,-2 & -3.

The sum would be then -6


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If x is an integer and |||x| - 3| - x| = 3, what is the sum of all pos 11 Feb 2021, 20:08
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Bunuel
If x is an integer and \(|||x| - 3| - x| = 3\), what is the sum of all possible values of x ?

A. -6
B. -4
C. -3
D. -1
E. 0

As also mentioned above, the solution for this would be \(x\geq{3}\) and \(0\geq{x}\geq{-3}\)

So sum will be (-3)+(-2)+(-1)+3+4+5....=Infinity


It is mentioned in the question that X is integer less tha equal to zero. Hence possible values of X can only be -1,-2 & -3.

The sum would be then -6


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If you read the thread again or the initial question to which this solution was given, you will realise that the question was edited later and the solution for the edited question is -6.
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