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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: 13 Sep 2018, 07:16
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Different methods to solve absolute value equations and inequalities- Exercise Question #1

If x is an integer, then how many values of x will satisfy the equation ||x - 2| + 7| = 6?

Options

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

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Originally posted by EgmatQuantExpert on 13 Sep 2018, 02:04.
Last edited by EgmatQuantExpert on 13 Sep 2018, 07:16, edited 2 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: 17 Sep 2018, 02:47
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1

Solution


Given:
    • We are given that x is an integer, and
    • We are also given an absolute value equation, ||x - 2| + 7| = 6

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

Approach and Working:
    • If we observe, we can see that there are two absolute value functions in the equation, ||x - 2| + 7| = 6
    • So, let us first solve the outer modulus by substituting |x - 2| as t, which gives
      o |t + 7| = 6
    • Now, applying the definition of |x|, as learnt in the article, we can write |t + 7| = 6 as,
      o t + 7 = 6, if t ≥ -7
         Implies, t = -1
         Can we consider this as a possible value of t?
         Obviously, No.
         Since, the value of t = |x - 2| is always greater than or equal to 0.
         Thus, t = -1 is not a possible value
      o And, t + 7 = -6, if t < -7
         This is again not possible, since, t cannot be a negative number
    • Thus, there is no value of t, which satisfies the equation, |t + 7| = 6, where t = |x - 2|
    • Therefore, the number of possible values of x is 0

Now, let’s see another approach to solve this equation.

    • In the equation, |t + 7| = 6, the minimum value of t = 0, since t = |x - 2|
    • So, the minimum value of |t + 7| = |0 + 7| = 7
    • Thus, the value of |t + 7| is always greater than or equal to 7
    • Therefore, the number of possible values of t and x is 0

Hence, the correct answer is option A.

Answer: A

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Originally posted by EgmatQuantExpert on 13 Sep 2018, 02:09.
Last edited by EgmatQuantExpert on 17 Sep 2018, 02:47, 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 13 Sep 2018, 07:24
EgmatQuantExpert wrote:
Different methods to solve absolute value equations and inequalities- Exercise Question #1

If x is an integer, then how many values of x will satisfy the equation ||x - 2| + 7| = 6?

Options

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

Next Question

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



Image


||x - 2| + 7| = 6
|x-2| will either be 0 or a +ve integer
0+7 <> 6
(+ve integer) + 7 <> 6

so 0

A
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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 13 Sep 2018, 08:08
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Left hand side to be equal to 6, x-2 must be equal to either -1 or -13 which is not possible because mod of x-2 always results in a positive value, so left hand side always will be equal to 7 or greater than 7 but can never be equal to 6.
So there is no value of x which can satisfy this equation.

Answer A

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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, 05:45
'A' it is, since solving the outer modulus we are getting negative values which is not possible.
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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 12 Oct 2018, 16:08
Can anyone explain this problem please? I dont understand the official solution.

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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 12 Oct 2018, 16:25
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Hey Jancekmichal --

The trick here is the idea that once you take the absolute value of something, it must be positive.

So based on that |x + 2| must be positive. This is important because of what follows.

Because an absolute value sign inside an absolute value sign is confusing, let's pretend that |x+2| = y instead. (This is just to simplify the equation. You could leave it as is)

That's going to give you | y + 7| = 6. Now solve that as an actual equation. Remember that with an absolute value sign you need to solve this twice, once as if what's inside the signs is negative and once as if it's positive. That will give you two equations:

y + 7 = 6, so y = -1 and y + 7 = -6, so y = - 13.

BUT WAIT. We said that y = |x + 2|, which we said must always be positive. That means that |x + 2| can't ever equal -6 or - 13, which means that there are no valid values for x that would make this equation true.
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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 12 Oct 2018, 16:29
Thanks so much, you are a life saver!

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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 Oct 2018, 22:49
If you have 2 absolute value expressions, you just need to consider 2 cases:

(1) both have the same sign
(2) they have different signs


Solving the equation using this approach you get 2 values for x:
x=1 and x=3.

Plugging them into the original equation, you see that both do not make the equation true.

Thus, the answer is A (0 solutions).

If you liked this approach, please hit Kudos! :)
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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 15 Oct 2018, 00:23
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EgmatQuantExpert wrote:
Different methods to solve absolute value equations and inequalities- Exercise Question #1

If x is an integer, then how many values of x will satisfy the equation ||x - 2| + 7| = 6?

Options

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




Given: ||x - 2| + 7| = 6

This is an inconsistent solution because
||x - 2| + 7| must be 7 or greater hence can NEVER be equal to 6

i.e. There is no possible solution for the expression

Answer: Option A

For similar question which can be solved, please check this question and our solutions

https://gmatclub.com/forum/how-many-dis ... l#p1543729

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If x is an integer, then how many values of x will satisfy the equatio   [#permalink] 15 Oct 2018, 00:23
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