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e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3074
If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 59% (01:35) correct 41% (01:44) wrong based on 308 sessions

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Different methods to solve absolute value equations and inequalities- Exercise Question #2

If x is an integer, then how many values of x will satisfy the equation ||x - 10| - 6| = 4?

Options

a) 0
b) 1
c) 2
d) 3
e) 4

To read the article: Different methods to solve absolute value equations and inequalities _________________

Originally posted by EgmatQuantExpert on 13 Sep 2018, 03:05.
Last edited by EgmatQuantExpert on 20 Sep 2018, 07:58, edited 5 times in total.
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3074
If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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Solution

Given:
• We are given that x is an integer, and
• We are also given an absolute value equation, ||x - 10| - 6| = 4

To find:
• We need to find the number of values of x, that satisfy the given equation

Approach and Working:
• As seen in the previous question, the first step is to substitute |x - 10| as t, which gives
o |t - 6| = 4
• Now, applying the definition of |x|, as learnt in the article, we can write |t - 6| = 4 as,
o t - 6 = 4, if t ≥ 6
 Implies, t = 10, which is greater than 6
 Now, substituting back t as |x - 10|, we get, |x - 10| = 10
 This again gives two cases,
Case -1: x – 10 = 10, if x ≥ 10
 Implies, x = 20, which is greater than 10
 Thus, x = 20, is a possible value of x
Case -2: x – 10 = -10, if x < 10
 Implies, x = 0, which is also a possible value
o And, t - 6 = -4, if t < 6
 Implies, t = 2, which is less than 6
 Now, substituting back t as |x - 10|, we get, |x - 10| = 2
 This again gives two cases,
Case -1: x – 10 = 2, if x ≥ 10
 Implies, x = 12, which is greater than 10
 Thus, x = 12, is a possible value of x
Case -2: x – 10 = -2, if x < 10
 Implies, x = 8, which is less than 10
 Thus, x = 8, is also a possible value
• Thus, the only possible values of x that satisfy the given equation are 0, 8, 12 and 20.
• Therefore, the number of possible values of x is 4.

Hence, the correct answer is option E.

_________________

Originally posted by EgmatQuantExpert on 13 Sep 2018, 03:10.
Last edited by EgmatQuantExpert on 20 Sep 2018, 21:06, edited 4 times in total.
Manager  G
Joined: 07 Aug 2018
Posts: 108
Location: United States (MA)
GMAT 1: 560 Q39 V28 GMAT 2: 670 Q48 V34 If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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1
My Approach:

Ix-10I will always be positive and in order for ||x-10|-6|=4 to be true |x-10|=2 or |x-10|=10

|x-10|=2

1.
x-10=2
x=12

2.
-x+10=2
x=8

--> x=12 or x=8

|x-10|=10

1.
x-10=20
x=20

2.
-x+10=10
x=0

Left with four values: 0,8,10,12

Please correct me if I am wrong!!

Edited! Was wrong first time...
_________________

Originally posted by T1101 on 13 Sep 2018, 10:36.
Last edited by T1101 on 20 Sep 2018, 08:05, edited 2 times in total.
Director  P
Joined: 31 Jul 2017
Posts: 512
Location: Malaysia
Schools: INSEAD Jan '19
GMAT 1: 700 Q50 V33 GPA: 3.95
WE: Consulting (Energy and Utilities)
Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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EgmatQuantExpert wrote:
Different methods to solve absolute value equations and inequalities- Exercise Question #2

If x is an integer, then how many values of x will satisfy the equation ||x - 10| - 6| = 4?

Options

a) 0
b) 1
c) 2
d) 3
e) 4

To read the article: Different methods to solve absolute value equations and inequalities Basically, here we have to find values of x for which |x-10| = 2,10.

So, x = 0,8,12,20

Hence, E.
_________________
If my Post helps you in Gaining Knowledge, Help me with KUDOS.. !!

Originally posted by rahul16singh28 on 13 Sep 2018, 10:46.
Last edited by rahul16singh28 on 14 Sep 2018, 04:41, edited 1 time in total.
Intern  B
Joined: 07 Jan 2018
Posts: 11
Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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1
x={0,8,12,20}
So isn't the answer E?

Posted from my mobile device
Intern  B
Joined: 08 Apr 2018
Posts: 16
Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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Answer would be 'E', 4 possible values of 'x' {0,8,12,20}.
Intern  B
Joined: 22 Apr 2017
Posts: 6
Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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EgmatQuantExpert wrote:

Solution

Given:
• We are given that x is an integer, and
• We are also given an absolute value equation, ||x - 10| - 6| = 4

To find:
• We need to find the number of values of x, that satisfy the given equation

Approach and Working:
• As seen in the previous question, the first step is to substitute |x - 10| as t, which gives
o |t - 6| = 4
• Now, applying the definition of |x|, as learnt in the article, we can write |t - 6| = 4 as,
o t - 6 = 4, if t ≥ 6
 Implies, t = 10, which is greater than 6
 Now, substituting back t as |x - 10|, we get, |x - 10| = 10
 This again gives two cases,
Case -1: x – 10 = 10, if x ≥ 10
 Implies, x = 20, which is greater than 10
 Thus, x = 20, is a possible value of x
Case -2: x – 10 = -10, if x < -10
 Implies, x = 0, which is not less than -10
 Thus, x = 0, is not a possible value
o And, t - 6 = -4, if t < 6
 Implies, t = 2, which is less than 6
 Now, substituting back t as |x - 10|, we get, |x - 10| = 2
 This again gives two cases,
Case -1: x – 10 = 2, if x ≥ 10
 Implies, x = 12, which is greater than 10
 Thus, x = 12, is a possible value of x
Case -2: x – 10 = -2, if x < 10
 Implies, x = 8, which is less than 10
 Thus, x = 8, is also a possible value
• Thus, the only possible values of x that satisfy the given equation are 8, 12 and 20.
• Therefore, the number of possible values of x is 3.

Hence, the correct answer is option D.

Hello Payal

In solution mentioned by you:
The case 2 of lx-10l=10; there should be -(x-10)=10 if x<10
So x=0 if x<10. Which is a possible solution.
So the answer should be E.
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3074
Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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Hey HimoGMAT, mittu, rahul16singh28, T1101

You all are correct.
There was a typo error.
We have corrected the solution.

Regards
_________________
Intern  B
Joined: 08 Apr 2018
Posts: 16
Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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EgmatQuantExpert wrote:
Hey HimoGMAT, mittu, rahul16singh28, T1101

You all are correct.
There was a typo error.
We have corrected the solution.

Regards

You also might want to update the second last line, where it currently says the number of possible solutions is 3.
Manager  B
Joined: 28 Jun 2018
Posts: 73
Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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chetan2u this one too please  Re: If x is an integer, then how many values of x will satisfy the equatio   [#permalink] 29 Sep 2018, 11:28
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