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Math Expert V
Joined: 02 Sep 2009
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For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

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Math Expert V
Joined: 02 Aug 2009
Posts: 8335
For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

Hi sobby,
The actual way would be...

Since both sides are positive, square both sides
$$(x-6)^2>(3x+6)^2..........x^2-12x+36>9x^2+36x+36.........8x^2+48x<0$$
8x(x+6)<0.....

For 8x(x+6) to be NEGATIVE, one of 8x and x+6 will be negative and other positive..

If x is positive, 8x will be positive and x+6 will also be positive..
So x should be NEGATIVE..
Then 8x will be negative, and thus x+6 should be positive ..
For that $$x+6>0.....X>-6....$$ But x<0.

So range becomes $$0>x>-6$$...
Values are $$-1,-2,-3,-4,-5$$
5 value
C
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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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7
3
Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

In this question, in my opinion, there are 2 possible solutions.

Solution 1.

$$|x – 6| > |3x + 6| \implies |x – 6| - |3x + 6|>0$$

If $$x<-2$$ we have $$(6-x)-(-3x-6)=6-x+3x+6=2x+12 > 0 \implies x >-6$$. Hence $$-6 <x <-2$$

If $$-2 \leq x < 6$$ we have $$(6-x)-(3x+6)=-4x>0 \implies x<0$$. Hence $$-2 \leq x < 0$$

If $$x \geq 6$$ we have $$(x-6)-(3x+6)=-2x-12>0 \implies x<-6$$. There is no satisfied value of $$x$$ in this case.

Combine all cases we have $$-6 < x < 0$$.

Thus, $$x$$ could receive 5 integer value $$\{-5;-4;-3;-2;-1\}$$

Solution 2.

$$|x – 6| > |3x + 6| \implies (x-6)^2 > (3x+6)^2$$

$$\implies x^2-12x+36 > 9x^2+36x+36 \implies 8x^2+48x<0$$

$$\implies 8x(x+6)<0 \implies -6<x<0$$.
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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

by substituting values i can see 5 as answer ...
No positive value for x can satisfy the equation...------------1
x=0 can equalize the equation-----------2
-1 to -5 will satisfy the equation...and -6 will equalize the condition-----3
so the answer is c - --5 ..

is there any other way to solve i'e without doing actual substitution ...
Thanks..
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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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sobby wrote:
Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

by substituting values i can see 5 as answer ...
No positive value for x can satisfy the equation...------------1
x=0 can equalize the equation-----------2
-1 to -5 will satisfy the equation...and -6 will equalize the condition-----3
so the answer is c - --5 ..

is there any other way to solve i'e without doing actual substitution ...
Thanks..

2nd approach.......

we can do it by squaring both side and using number line to solve it ...
we will get the roots as 0 and -6 and using this we can deduce only number between 0 and -6 will satisfy the actual solution..

Bunuel : Is this approach correct

I have one doubt ---while squaring in absolute ....what necessary condition we need to consider....
Math Expert V
Joined: 02 Sep 2009
Posts: 60627
Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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sobby wrote:
sobby wrote:
Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

by substituting values i can see 5 as answer ...
No positive value for x can satisfy the equation...------------1
x=0 can equalize the equation-----------2
-1 to -5 will satisfy the equation...and -6 will equalize the condition-----3
so the answer is c - --5 ..

is there any other way to solve i'e without doing actual substitution ...
Thanks..

2nd approach.......

we can do it by squaring both side and using number line to solve it ...
we will get the roots as 0 and -6 and using this we can deduce only number between 0 and -6 will satisfy the actual solution..

Bunuel : Is this approach correct

I have one doubt ---while squaring in absolute ....what necessary condition we need to consider....

We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality). So, if you have absolute values on both sides of the inequality then you can safely square.

Check for more the following topics:

Solving Quadratic Inequalities - Graphic Approach
Inequality tips

DS Inequalities Problems
PS Inequalities Problems

700+ Inequalities problems

Hope it helps.
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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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1
chetan2u wrote:
Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

Hi sobby,
The actual way would be...

Since both sides are positive, square both sides
$$(x-6)^2>(3x+6)^2..........x^2-12x+36>9x^2+36x+36.........8x^2+36x<0$$
8x(x+6)<0.....
For 8x(x+6) to be NEGATIVE, one of 8x and x+6 will be negative and other positive..
If x is positive, 8x will be positive and x+6 will also be positive..
So x should be NEGATIVE..
Then 8x will be negative, and thus x+6 should be positive ..
For that x+6>0.....X>-6.... But x<0.
So range becomes 0>x>-6...
Values are -1,-2,-3,-4,-5
5 value
C

Attachments

File comment: Here is another graphical approach.

The blue graph represents |x-6| and red graph represents |3x+6| and these two graphs meet each other at x=0 and x=-6
For |x-6| to be greater than |3x+6| the blue graph should be above red graph. So our solution should be -6<x<0 i.e. -1,-2,-3,-4,-5.

Experts please comment if this approach is correct. Also I tried applying the same approach to 3 modulus equations but it is not so handy as it is here.. Please suggest some method for 3 modulus problems as |x – 8| + |5 – x| > |x + 7| Graph1.PNG [ 34.52 KiB | Viewed 10582 times ]

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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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3
2
Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

Think in terms of the definition of absolute values:

|x - 6| > 3*|x + 2|

Distance from 6 is greater than 3 times the distance from -2

-------- (-2) -------- (0) ------------------------------ (6) -------------

At point 0, the distance from 6 will be equal to 3 times the distance from -2. So on the left of 0, the distance from 6 will be greater.

The distance between -2 and 6 is 8. Distance from 6 will be equal to 3 times the distance from -2 at a point 4 units to the left of -2 i.e. -6. To its left, distance from 6 will be less. So between -6 to 0, distance from 6 is greater than 3 times the distance from -2. We have 5 integer values lying between these two.

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For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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10
3
Bunuel wrote:
For how many integer values of x, is $$|x – 6| > |3x + 6|$$?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

To solve an inequality where the absolute value of an expression is greater than (or less than) the absolute value of another expression, we need to find the values that make the absolute values of the two expressions equal to each other. That is, to solve |x – 6| > |3x + 6|, we first need to solve |x – 6| = |3x + 6|. Recall that if |A| = |B|, then either A = B (Case 1) or A = -B (Case 2).

Case 1: When $$x - 6 = 3x + 6$$:

$$x - 6 = 3x + 6$$

$$-2x = 12$$

$$x = -6$$

Case 2: When $$x - 6 = -(3x + 6)$$:

$$x - 6 = -(3x + 6)$$

$$x - 6 = -3x - 6$$

$$4x = 0$$

$$x = 0$$

These two values, -6 and 0, are the critical values for determining the solution to the given inequality. Now when we place these two values on the number line, we will see that they partition the number line into 3 intervals:

$$x < -6$$

$$-6 < x < 0$$

$$x > 0$$

For each of these intervals, we will pick a value in that interval (for example, for x < -6, we can pick -7) and test whether it satisfies the given inequality. If it does, then EVERY number in that interval will satisfy the inequality. If it doesn’t, then NO numbers in that interval will satisfy the inequality. Recall that our given inequality is |x – 6| > |3x + 6|. Let’s begin testing the intervals.

1) For $$x < -6$$, let’s pick x = -7:

|-7 – 6| > |3(-7) + 6| ?

$$|-13| > |-15|$$?

$$13 > 15$$ → No!

Thus, no numbers in the interval $$x < -6$$will satisfy $$|x – 6| > |3x + 6|.$$

2) For $$-6 < x < 0$$, let’s pick x = -1:

$$|-1 – 6| > |3(-1) + 6|$$?

$$|-7| > |3|$$ ?

$$7 > 3$$→ Yes

Thus, every number in the interval $$-6 < x < 0$$ will satisfy $$|x – 6| > |3x + 6|.$$

3) For $$x > 0$$, let’s pick x = 1:

$$|1 – 6| > |3(1) + 6|$$ ?

$$|-5| > |9|$$ ?

$$5 > 9$$ → No

Thus, no numbers in the interval $$x > 0$$ will satisfy $$|x – 6| > |3x + 6|.$$As we can see, the only interval that satisfies $$|x – 6| > |3x + 6|$$ is $$-6 < x < 0$$. In other words, the solution to $$|x – 6| > |3x + 6|$$ is $$6 < x < 0$$. In this interval, there are 5 integers: -5, -4, -3, -2, and -1.

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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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VeritasPrepKarishma wrote:
Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

Think in terms of the definition of absolute values:

|x - 6| > 3*|x + 2|

Distance from 6 is greater than 3 times the distance from -2

-------- (-2) -------- (0) ------------------------------ (6) -------------

At point 0, the distance from 6 will be equal to 3 times the distance from -2. So on the left of 0, the distance from 6 will be greater.

The distance between -2 and 6 is 8. Distance from 6 will be equal to 3 times the distance from -2 at a point 4 units to the left of -2 i.e. -6. To its left, distance from 6 will be less. So between -6 to 0, distance from 6 is greater than 3 times the distance from -2. We have 5 integer values lying between these two.

Responding to a pm:
Quote:
I know very well that if |x - 3| < 6

Distance of x from 3 is less than 6 so -3 < x < 9.

First consider the equation: |x - 6| = 3*|x + 2|
That is, the point where distance from 6 is equal to three times the distance from -2. There is such a point between 6 and -2. That point is x = 0. Here the distance from 6 is 6 which is equal to three times 2, the distance of 0 from -2.
How do we get x = 0?
Distance from 6 should be thrice the distance from -2
Distance from 6 : Distance from -2 = 3:1
Split the distance between 6 and -2 into the ratio 3:1. Since there are 8 units between 6 and -2, you get that x is 6 units away from 6 and hence at 0.

For the inequality, we want the distance from 6 to be greater than the distance from -2 so we move to the left (closer to -2). The distance from 6 will keep increasing till we reach -2 when the distance from 6 is 8. Note that now also distance from 6 is greater than 3 times the distance from -2.

Now as we go further to the left, the distance from 6 keeps increasing but so does the distance from -2. There will be some point again where distance from 6 will be equal to 3 times the distance from -2. At this point, the distance of 8 units between 6 and -2 will make up twice the distance of x from -2

(x)......................(-2).................(0)...................................................(6)...................

< Distance from -2>

<-------------------Distance from 6 ----------------------->

So x must be 4 units to the left of -2. So at x = -6, again, distance from 6 will be three times the distance from -2.

Hence, for our inequality, x can take values between -6 and 0. This gives us 5 integer values.
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For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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VeritasPrepKarishma wrote:
Responding to a pm:
Quote:
I know very well that if |x - 3| < 6

Distance of x from 3 is less than 6 so -3 < x < 9.

First consider the equation: |x - 6| = 3*|x + 2|
That is, the point where distance from 6 is equal to three times the distance from -2. There is such a point between 6 and -2. That point is x = 0. Here the distance from 6 is 6 which is equal to three times 2, the distance of 0 from -2.
How do we get x = 0?
Distance from 6 should be thrice the distance from -2
Distance from 6 : Distance from -2 = 3:1
Split the distance between 6 and -2 into the ratio 3:1. Since there are 8 units between 6 and -2, you get that x is 6 units away from 6 and hence at 0.

For the inequality, we want the distance from 6 to be greater than the distance from -2 so we move to the left (closer to -2). The distance from 6 will keep increasing till we reach -2 when the distance from 6 is 8. Note that now also distance from 6 is greater than 3 times the distance from -2.

Now as we go further to the left, the distance from 6 keeps increasing but so does the distance from -2. There will be some point again where distance from 6 will be equal to 3 times the distance from -2. At this point, the distance of 8 units between 6 and -2 will make up twice the distance of x from -2

(x)......................(-2).................(0)...................................................(6)...................

< Distance from -2>

<-------------------Distance from 6 ----------------------->

So x must be 4 units to the left of -2. So at x = -6, again, distance from 6 will be three times the distance from -2.

Hence, for our inequality, x can take values between -6 and 0. This gives us 5 integer values.

Thank you so much VeritasPrepKarishma for the explanation. +1

A good thoughtful explanation. It took a little time for me to understand but I guess I need to practice more to internalize it.

Just one question here, for the below inequality -
|x-6| > |x+2|
can we say that the solution for this is x<2?
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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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4
RMD007 wrote:
VeritasPrepKarishma wrote:
Responding to a pm:
Quote:
I know very well that if |x - 3| < 6

Distance of x from 3 is less than 6 so -3 < x < 9.

First consider the equation: |x - 6| = 3*|x + 2|
That is, the point where distance from 6 is equal to three times the distance from -2. There is such a point between 6 and -2. That point is x = 0. Here the distance from 6 is 6 which is equal to three times 2, the distance of 0 from -2.
How do we get x = 0?
Distance from 6 should be thrice the distance from -2
Distance from 6 : Distance from -2 = 3:1
Split the distance between 6 and -2 into the ratio 3:1. Since there are 8 units between 6 and -2, you get that x is 6 units away from 6 and hence at 0.

For the inequality, we want the distance from 6 to be greater than the distance from -2 so we move to the left (closer to -2). The distance from 6 will keep increasing till we reach -2 when the distance from 6 is 8. Note that now also distance from 6 is greater than 3 times the distance from -2.

Now as we go further to the left, the distance from 6 keeps increasing but so does the distance from -2. There will be some point again where distance from 6 will be equal to 3 times the distance from -2. At this point, the distance of 8 units between 6 and -2 will make up twice the distance of x from -2

(x)......................(-2).................(0)...................................................(6)...................

< Distance from -2>

<-------------------Distance from 6 ----------------------->

So x must be 4 units to the left of -2. So at x = -6, again, distance from 6 will be three times the distance from -2.

Hence, for our inequality, x can take values between -6 and 0. This gives us 5 integer values.

Thank you so much VeritasPrepKarishma for the explanation. +1

A good thoughtful explanation. It took a little time for me to understand but I guess I need to practice more to internalize it.

Just one question here, for the below inequality -
|x-6| > |x+2|
can we say that the solution for this is x<2?

Yes, that is correct.

|x-6| > |x+2|

The distance of x from 6 is more than the distance of x from -2.

The two distances will be equal at the mid point that is at x = 2. As you move the left, the distance of x from 6 will keep increasing. To the left of -2 too, the distance from 6 will always be more than the distance from -2.

Hence answer will be x < 2.
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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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In this question, in my opinion, there are 2 possible solutions.

Solution 1.

|x–6|>|3x+6|⟹|x–6|−|3x+6|>0|x–6|>|3x+6|⟹|x–6|−|3x+6|>0

If x<−2x<−2 we have (6−x)−(−3x−6)=6−x+3x+6=2x+12>0⟹x>−6(6−x)−(−3x−6)=6−x+3x+6=2x+12>0⟹x>−6. Hence −6<x<−2−6<x<−2

If −2≤x<6−2≤x<6 we have (6−x)−(3x+6)=−4x>0⟹x<0(6−x)−(3x+6)=−4x>0⟹x<0. Hence −2≤x<0−2≤x<0

If x≥6x≥6 we have (x−6)−(3x+6)=−2x−12>0⟹x<−6(x−6)−(3x+6)=−2x−12>0⟹x<−6. There is no satisfied value of xx in this case.

Combine all cases we have −6<x<0−6<x<0.

Thus, xx could receive 5 integer value {−5;−4;−3;−2;−1}{−5;−4;−3;−2;−1}

Solution 2.

|x–6|>|3x+6|⟹(x−6)2>(3x+6)2|x–6|>|3x+6|⟹(x−6)2>(3x+6)2

⟹x2−12x+36>9x2+36x+36⟹8x2+48x<0⟹x2−12x+36>9x2+36x+36⟹8x2+48x<0

⟹8x(x+6)<0⟹−6<x<0⟹8x(x+6)<0⟹−6<x<0.

hi nguyendinhtuong .
could u please explain why x<-6 has no satisfied values in solution 1.

thank you..
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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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3
Hi All,

One of the great 'design aspects' of most GMAT questions is that they can be approached in a variety of ways. As such, just because you got a question correct doesn't necessarily mean that you solved it in the fastest/easiest way possible (and if you're doing a lot of complex math to get to the solution, then you are likely NOT using the fastest approach). Sometimes the simplest approach to certain questions is just 'brute force' arithmetic - plow through the basic math necessary to prove the solution.

Here, we're asked for the number of INTEGER solutions to |X - 6| > |3X + 6|.

From the answer choices, it seems clear that there aren't that many possibilities (and it's unlikely that there's an 'unlimited' number of solutions), so we just have to find the 1, 3, 5 or 7 options that 'fit' this inequality.

To start, let's TEST a couple of simple values for X: 0 and 1....

IF...
X = 0, then we end up with...
|-6| > |6| ..... 6 > 6.... which is NOT correct - so X=0 is NOT a solution

IF...
X = 1, then we end up with...
|-5| > |9| ..... 5 > 9.... which is NOT correct - so X=1 is NOT a solution

Increasing the value of X will just make the "right side' absolute value a lot bigger, so there's no point in raising X. Thus, let's decrease it and see what happens....

IF...
X = -1, then we end up with...
|-7| > |3| ..... 7 > 3.... which IS correct - so X=-1 IS a solution

At this point, how long would it take you to TEST -2, -3, -4, -5 and -6? What happens when you try -6 though? And what about -7? Notice how you would stop working for the same reason why you wouldn't both trying 2, 3, 4, etc... Now, how many total solutions do you have?

GMAT assassins aren't born, they're made,
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For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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Bunuel wrote:
For how many integer values of x, is $$|x – 6| > |3x + 6|$$ ?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

Bunuel Why my solution is incorrect?

(1) Take positive

$$|x – 6|=x - 6$$
$$|3x + 6|=3x + 6$$

$$x-6>3x+6$$
$$x<-6$$

(2) Take negative

$$|x-6| = -x+6$$
$$|3x+6| = -3x-6$$

$$-x+6>-3x-6$$
$$x>-6$$

Does this approach correct? https://gmatclub.com/forum/for-how-many ... l#p1804531
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GRE 1: Q169 V154 Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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ziyuen wrote:
Bunuel wrote:
For how many integer values of x, is $$|x – 6| > |3x + 6|$$ ?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

Bunuel Why my solution is incorrect?

(1) Take positive

$$|x – 6|=x - 6$$
$$|3x + 6|=3x + 6$$

$$x-6>3x+6$$
$$x<-6$$

(2) Take negative

$$|x-6| = -x+6$$
$$|3x+6| = -3x-6$$

$$-x+6>-3x-6$$
$$x>-6$$

Does this approach correct? https://gmatclub.com/forum/for-how-many ... l#p1804531

Hi ziyuen
It is not that easy.
This is an extremely difficult problem.
You highlighted part is incorrect.

Here is what did =>

Firstly the definition of |x| -->
|x|= x for x>0
= -x for x<0
=0 for x=0

Now lets look at the two modulus equations -->
|x-6| = x-6 for x>6
=-(x-6)=-x+6 for x<6
=0 for x=6

Similarly
|3x+6| => 3x+6 for x>-2
=> -3x-6 for x<-2
=> 0 for x=-2

Hence the three boundaries are => (-∞,-2] , [-2,6]and [6,∞)

For x>6 ==>

x-6>3x+6
x<-6

REJECTED as x cannot be x>6 and x<-6 simultaneously.
((Also x≠6 as modulus is always positive))

For x=> [-2,6)
-x+6>3x+6
4x4x<0
x<0

Hence x can be -2 or -1

Finally when x<-2 ==>
-x+6 >-3x-6
2x>-12
x>-6

Hence x can be -5,-4,-3

Final conclusion --> x can be -5,-4,-3,-2,-1

FIVE VALUES.

SMASH THAT C.

P.S-> It took me about 3 minutes and 32 seconds to get to the correct answer.
This question is extremely helpful to clear basics.

Also you can go through
chetan2u 's blog here --> http://gmatclub.com/forum/absolute-modu ... l#p1622372

Regards
Stone Cold

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For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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Bunuel wrote:
For how many integer values of x, is |x – 6| > |3x + 6|?

(A) 1

(B) 3

(C) 5

(D) 7

(E) Infinite

MY way simple and sober and within a minute you get the answer

equate x – 6 = 3x + 6
we get x=-6
Now if you will decrease x let's say
Quote:
-7 ((try it)
then |x – 6| > |3x + 6| wil not hold ((LHS WILL BE LESS THAN RHS

However if you DECREASE X
Quote:
say -5
then |x – 6| > |3x + 6| holds
this continues till -1 and then again fails
therefore -5.-4,-3,-2,-1 check for 0
Ans =5
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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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|x-6| > |3x + 6|?

(X - 6)^2 > (3x + 6)^2
x^2 - 12x + 36 > 9x^2 + 36x + 36
8x^2 + 48x < 0
8x ( x + 6) < 0

Now, for it to be negative, either 1) 8x is positive and (x+6) is negative or 2) 8x is negative and (x+6) is positive. Both cannot be positive and both cannot be negative

1) For 8x to be positive and x+6 to be negative is impossible. As you cannot have a positive number x that when multiplied by 8 is positive and when added to 6 is negative, then no results here.
2) For 8x is negative and (x+6) is positive: Any negative number multiplied by 8 is negative, but not any negative number added to 6 is positive.
The only negative integers that when added to 6 become positive are -1, -2, -3, -4 and -5

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Re: For how many integer values of x, is |x – 6| > |3x + 6|?  [#permalink]

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By far the easiest approach to these super rare, super hard inequality problems is number plugging

0: NO
1: NO
2: NO
.
.
.
100000:NO

-1: YES
-2: YES
-3: YES
-4: YES
-5: YES
-6: NO
.
.
.
-100000: NO

done (t=45sec)
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Good luck to you. Retired from this forum. Re: For how many integer values of x, is |x – 6| > |3x + 6|?   [#permalink] 09 Nov 2019, 00:33
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