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

Question

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

https://e-gmat.com/blogs/wp-content/uploads/2018/08/Free-Session-Algebra-e1533292089856.jpg

T1101 · Accepted Answer

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...

EgmatQuantExpert · Answer

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

https://e-gmat.com/blogs/wp-content/uploads/2018/08/Free-Session-Algebra-e1533292089856.jpg

rahul16singh28 · Answer

Basically, here we have to find values of x for which |x-10| = 2,10.

So, x = 0,8,12,20

Hence, E.

mittu · Answer

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

Posted from my mobile device

sayankh · Answer

Answer would be 'E', 4 possible values of 'x' {0,8,12,20}.

HimoGMAT · Answer

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.

EgmatQuantExpert · Answer

Hey  HimoGMAT ,  mittu ,  rahul16singh28 ,  T1101

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

Regards

sayankh · Answer

You also might want to update the second last line, where it currently says the number of possible solutions is 3.

hibobotamuss · Answer

chetan2u this one too please

Ved22 · Answer

The best approach to solve this problem within 1 min is to plot the equation as a graph in coordinate axis.

bumpbot · Answer

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

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