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How many integer solutions are there to the inequality |x+8|<x?

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Economist GMAT Tutor Representative
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How many integer solutions are there to the inequality |x+8|<x?  [#permalink]

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New post 13 Oct 2019, 01:06
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How many integer solutions are there to the inequality \(|x+8|<x\)?

A. 0

B. 2

C. 3

D. 8

E. 16

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Re: How many integer solutions are there to the inequality |x+8|<x?  [#permalink]

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New post 13 Oct 2019, 03:18
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TheEconomistGMAT wrote:
How many integer solutions are there to the inequality \(|x+8|<x\)?

A. 0

B. 2

C. 3

D. 8

E. 16


x can never take negative values as Modulus |x+8| can never be negative

Also, when x is 0 or positive |x+8| is always greater than x

So, no values of x satisfies the above inequality

IMO Option A
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Re: How many integer solutions are there to the inequality |x+8|<x?  [#permalink]

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New post 14 Oct 2019, 07:59
1) if, X +8 <0, -(x+8)<x, ==> X <-8 or x >-4, so no integer solutions.
2) if, X +8 >0, (x+8)<x, ==> X so no integer solutions.

Answer A.
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Re: How many integer solutions are there to the inequality |x+8|<x?  [#permalink]

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New post 15 Oct 2019, 22:03
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Official Explanation



The issue in this question is an inequality combined with an absolute value. Note that this inequality is of the variable form (i.e., variables on both sides). Thus, the two-scenario approach will not necessarily work. Instead, plug in numbers to find out what the inequality really means.

First note that an absolute value can never be less than zero or a negative number, so x must be positive. Try some positive values of xx and see what happens.

Plug in x=2
|x+8|=|2+8|=|10|=10
This is NOT smaller than x=2 Therefore, x cannot equal 2.

Try other positive values of x Plug in x=3
|x+8|=|3+8|=|11|=11
This is NOT smaller than x=3. Therefore, x cannot equal 3.

Plug in x=0.5
|x+8|=|0.5+8|=|8.5|=8.5
This is NOT smaller than x=0.5 Therefore, x cannot equal 0.5.

Notice a pattern? For every positive value of x, the left side of the equation will always be greater than the right side. Thus, |x+8||x+8| will never be smaller than x, and there are no values of x that satisfy the inequality.

Answer: A
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Re: How many integer solutions are there to the inequality |x+8|<x?   [#permalink] 15 Oct 2019, 22:03
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