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For How many integer values of x, is |2x-5|+|x+1|+|x|<10?

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For How many integer values of x, is |2x-5|+|x+1|+|x|<10? [#permalink]

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New post 13 Feb 2017, 06:29
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For how many integer values of x, is \(|2x-5|+|x+1|+|x|<10\)?

A. 1
B. 2
C. 4
D. 5
E. Infinite

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Re: For How many integer values of x, is |2x-5|+|x+1|+|x|<10? [#permalink]

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New post 13 Feb 2017, 06:59
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5
ziyuenlau wrote:
For How many integer values of x, is \(|2x-5|+|x+1|+|x|<10\)?

A. \(1\)
B. \(2\)
C. \(4\)
D. \(5\)
E. \(Infinite\)


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Re: For How many integer values of x, is |2x-5|+|x+1|+|x|<10? [#permalink]

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New post 13 Feb 2017, 22:21
Hi ziyuenlau,

We're told that X must be an INTEGER, so since we're adding the sums of 3 absolute values - AND we're looking for sums that are LESS than 10 - there cannot be an infinite number of possibilities. Thus, based on the answer choices, there can't be more than 5 possible integer values for N. As such, we can use a bit of 'brute force' to find all of the possible values without too much trouble.

We're given the inequality: |2X - 5| + |X + 1| + |X| < 10.

Let's start with...
X = 0.... |-5| + |1| + |0| = 6 so X could be 0

Now let's work our way "up"...
X = 1.... |-3| + |2| + |1| = 6 so X could be 1
X = 2.... |-1| + |3| + |2| = 6 so X could be 2
X = 3.... |1| + |4| + |3| = 8 so X could be 3
X = 4.... |3| + |5| + |3| = 11 so X CANNOT be 4

Now let's work our way "down"....
X = -1.... |-7| + |0| + |-1| = 8 so X could be -1

At this point, we have 5 possible values of X; since that it the largest of the possible values, we can stop working.

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Re: For How many integer values of x, is |2x-5|+|x+1|+|x|<10? [#permalink]

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New post 02 Jan 2018, 01:45
Bunuel mikemcgarry MathRevolution: Is the below method correct to solve the given type of inequality questions ? Is this another correct way instead of considering the individual ranges ?

1. Taking all positive terms :
2x-5+x+1+x<10
4x-4<10
x<7/2

2. Taking all negative terms :
5-2x-x-x-x<10
x>-3/2

So the range is -3/2<x<7/2

Therefore , the possible values are -1,0,1,2,3...so 5 values
Re: For How many integer values of x, is |2x-5|+|x+1|+|x|<10?   [#permalink] 02 Jan 2018, 01:45
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