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

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

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New post Updated on: 20 Sep 2018, 07:58
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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

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

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

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New post 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.

Answer: E

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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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New post 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
So Answer: E



Please correct me if I am wrong!!

Edited! Was wrong first time...
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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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Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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New post 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

Previous Question | Next Question


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

Image


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

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New post 14 Sep 2018, 04:39
1
x={0,8,12,20}
So isn't the answer E?

Posted from my mobile device
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Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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New post 14 Sep 2018, 07:28
Answer would be 'E', 4 possible values of 'x' {0,8,12,20}.
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Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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New post 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.

Answer: D

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

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New post 20 Sep 2018, 08:03
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Re: If x is an integer, then how many values of x will satisfy the equatio  [#permalink]

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

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New post 29 Sep 2018, 11:28
chetan2u this one too please :(
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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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