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# If x is an integer, then how many values of x will satisfy the equatio

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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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Updated on: 20 Sep 2018, 07:58
00:00

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.
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Posts: 3074
If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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Updated on: 20 Sep 2018, 21:06

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.
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If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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Updated on: 20 Sep 2018, 08:05
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.
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Joined: 31 Jul 2017
Posts: 512
Location: Malaysia
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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Updated on: 14 Sep 2018, 04:41
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.
_________________
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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
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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14 Sep 2018, 04:39
1
x={0,8,12,20}

Posted from my mobile device
Intern
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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14 Sep 2018, 07:28
Answer would be 'E', 4 possible values of 'x' {0,8,12,20}.
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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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20 Sep 2018, 07:45
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.
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Posts: 3074
Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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20 Sep 2018, 08:03
Intern
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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20 Sep 2018, 08:44
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
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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29 Sep 2018, 11:28
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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