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For how many integer values of x, is |3x-3|+|2x+8|<15?

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For how many integer values of x, is |3x-3|+|2x+8|<15? [#permalink]

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For how many integer values of x, is \(|3x-3|+|2x+8|<15\)?

A. 2
B. 3
C. 4
D. 5
E. 6
[Reveal] Spoiler: OA

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Last edited by hazelnut on 13 Feb 2017, 06:20, edited 1 time in total.

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Re: For how many integer values of x, is |3x-3|+|2x+8|<15? [#permalink]

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ziyuenlau wrote:
How to resolve this question?

For how many integer values of x, is \(|3x-3|+|2x+8|<15\)?



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Re: For how many integer values of x, is |3x-3|+|2x+8|<15? [#permalink]

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we can solve this by following ways -

1. First take the positive parts of mod values -

|3x-3| = 3x-3
|2x+8| = 2x+8

So, it boils down to -

3x-3+2x+8 < 15
x<2

2. Now consider the negative parts of them -

|3x-3| = -3x+3
|2x+8| = -2x-8

-3x+3-2x-8<15
-x<4
x>-4

So , we can conclude that: -4<x<2
Thus, we will have - 5 integers, which will satisfy this equation.

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Re: For how many integer values of x, is |3x-3|+|2x+8|<15? [#permalink]

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New post 13 Feb 2017, 06:11
ziyuenlau wrote:
How to resolve this question?

For how many integer values of x, is \(|3x-3|+|2x+8|<15\)?



ziyuenlau

Please provide answer choices.

Answer must be 5 ...please confirm

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Re: For how many integer values of x, is |3x-3|+|2x+8|<15? [#permalink]

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New post 13 Feb 2017, 07:16
\(3x-3 = 0\) then \(x = 1\)
\(2x+8 = 0\) then \(x = -4\)

there can be 3 range
\(x\leq{-4}\) | \(-4 < x < 1\) | \(x\geq{1}\)

1. When \(x\leq{-4}\), then both \(|3x-3|\) and \(|2x+8|\) will be negative.

\(-3x+3 -2x-8 < 15\)

\(-5x-5 < 15\)

\(x > -4\) (this is opposite to \(x\leq{-4}\). )

Hence \(x\leq{-4}\) is not a possibility.

2. When \(-4 < x < 1\), then \(|3x-3|\) would still be negative but \(|2x+8|\) will be positive. hence

\(-3x+3 +2x+8 < 15\)

-x+11 < 15

\(-x < 4\) i.e. \(x > -4\) (this lies within \(-4 < x < 1\)) so correct range.

3. When \(x\geq{1}\), then both \(|3x-3|\) and \(|2x+8|\) will be positive.

\(3x-3+2x+8 < 15\)

\(5x+5 < 15\)

\(x+1 < 3\)

\(x < 2\) this range is also fine because \(x\geq{1}\). so x must be 1.

So our range is \(-4 <\)\(x\geq{1}\), so howmany integers within this range?

-3, -2, -1, 0 and 1

Count is 5

Answer D.
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Re: For how many integer values of x, is |3x-3|+|2x+8|<15? [#permalink]

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ziyuenlau wrote:
For how many integer values of x, is \(|3x-3|+|2x+8|<15\)?

A. 2
B. 3
C. 4
D. 5
E. 6


\(|3x-3|+|2x+8|<15\)

\(3*|x-1|+2*|x+4|<15\)

We want the values of x such that the sum of "thrice their distance from 1" and "twice their distance from -4" is less than 15.

Let's try to find the point where this distance is equal to 15.

........................ (-4) ...................................... (0) ........... (1) ..........................

The distance between -4 and 1 is 5. Thrice this distance is 15. So at the point x = -4, the sum will be 15. As we move to the right of -4, the sum will reduce (since the twice component will keep increasing). At x = 1, the sum becomes 0 + 2*5 = 10.
What happens when you go to the right of 1? Now the sum starts increasing since the thrice components increasing now.
At x = 2, the sum becomes 3*1 + 2*6 = 15.
To the right of 2, the sum will keep increasing.

So the sum will be less than 15 between -4 and 2. This gives us 5 integer values (-3, -2, -1, 0, 1).

Answer (D)
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Re: For how many integer values of x, is |3x-3|+|2x+8|<15? [#permalink]

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Re: For how many integer values of x, is |3x-3|+|2x+8|<15? [#permalink]

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Buttercup3 wrote:


9. Inequalities



For more check Ultimate GMAT Quantitative Megathread



Hope it helps.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: For how many integer values of x, is |3x-3|+|2x+8|<15?   [#permalink] 01 Nov 2017, 20:09
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