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Director  Joined: 03 Sep 2006
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What is the value of integer x? (M27-14)  [#permalink]

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Question Stats: 27% (02:12) correct 73% (02:21) wrong based on 311 sessions

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What is the value of integer x?

(1) 2x^2+9<9x
(2) |x+10|=2x+8
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Re: What is the value of integer x?  [#permalink]

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4
9
LM wrote:
What is the value of integer x?

(1) 2x^2+9<9x
(2) |x+10|=2x+8

What is the value of integer x?

(1) 2x^2+9<9x --> factor qudratics: $$(x-\frac{3}{2})(x-3)<0$$ --> roots are $$\frac{3}{2}$$ and 3 --> "<" sign indicates that the solution lies between the roots: $$1.5<x<3$$ --> since there only integer in this range is 2 then $$x=2$$. Sufficient.

(2) |x+10|=2x+8 --> LHS is an absolute value, which is always non negative, hence RHS must also be non-negative: $$2x+8\geq{0}$$ --> $$x\geq{-4}$$, for this range $$x+10$$ is positive hence $$|x+10|=x+10$$ --> $$x+10=2x+8$$ --> $$x=2$$. Sufficient.

Check this for more on solving inequalities like the one in the first statement:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.
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Re: What is the value of integer x?  [#permalink]

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I understand that statement 1 is sufficient....bt
Statement 2, i tried solving and gt two different answers....

lx+10l = 2x+8
when x>0
x+10 =2x+8
x=2
Another case when x<0
-(x+10) =2x+8
x=-6

Am i wrong while solving the mod....

Help me with statement 2

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Re: What is the value of integer x?  [#permalink]

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Hi Archit143,

Lets see statement 2: (2) |x+10|=2x+8

We have to analyze the two cases so
1)Range: $$x+10>0, x>-10$$
$$x+10=2x+8$$
$$x=2$$
$$-2$$ is in the range we are analyzing (>-10) so -2 is a solution

2)Range: $$x+10<0,x<-10$$
$$-x-10=2x+8$$
$$x=-6$$
$$-6$$ is OUT of the range we are considering (<-10) so is NOT a possible solution

Let me know if it's clear
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Re: What is the value of integer x?  [#permalink]

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thnks that was subtle i frgot to check the range.....
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Re: M27-14 What is the value of integer x?  [#permalink]

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1
HumptyDumpty wrote:
My way:
standard procedure:
1st: $$x+10=2x+8$$
$$x=2$$
OR
2nd:$$-(x+10)=2x+8$$
$$x=-6$$
the condition $$x=i$$ is met:
Insufficient.

This approach is NOT correct.

Always be sure to plug back the solutions for equations with modulus sign
the values that you have found are x = 2 and x = -6
ONLY x=2 satisfies the given condition
For X = -6

LHS = |x+10| = 4
RHS = 2x + 8 = -4
SO x = -6 is NO GOOD

This is happening because you have found the roots of
$$(x+10)^2$$ = $$(2x+8)^2$$
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Re: M27-14 What is the value of integer x?  [#permalink]

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This could ALSO be solved YOUR way but you have missed the definition of the modulus sign
|x|= x WHEN x >or = 0
|x| = -x WHEN x<0

Now lets come back to the problem,

|x+10| = 2x+8

SO x+10 = 2x+8 when x+10> 0
So it gives x=2
Now lets check: is 2+10 > 0 YES. So this is GOOD
Next
-(x+10) = 2x+8 when x+10<0
it gives x=-6
Lets check: is -6+10 < 0 NO. So this is what you call an extraneous root. Does no good.
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Re: M27-14 What is the value of integer x?  [#permalink]

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Is this approach completely NOT correct,
or
is this approach just missing the check-step?

What's the earliest stage in solving in which the exclusion can be spotted (by a not-so-pro fellow)?
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Re: M27-14 What is the value of integer x?  [#permalink]

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1
HumptyDumpty wrote:
Is this approach completely NOT correct,
or
is this approach just missing the check-step?

What's the earliest stage in solving in which the exclusion can be spotted (by a not-so-pro fellow)?

Well you just need to know the definition of modulus

sure |x| = 8 will give you 8 and -8 as solution but that will not hold true when the RHS also has a variable

when both sides consist variable go by the definition.

You have only taken a part of the definition. You need to look at the entire scene as explained in the second post.
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Re: M27-14 What is the value of integer x?  [#permalink]

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HumptyDumpty wrote:
What is the value of integer x?

1) $$2x^2+9<9x$$
2) $$|x+10|=2x+8$$

1) is clear: compute the quadratic expression, consider the scope of values and the condition x=i:

$$2x^2-9x+9<0$$
$$x^2-\frac{9x}{2}+\frac{9}{2}<0$$
$$(x-3)(x-\frac{3}{2})<0$$

the sums in parentheses must have opposite signs, so:

$$\frac{3}{2}<x<3$$

consider the condition $$x=i$$:

$$x=2$$
Sufficient.

2) is not clear:

$$|x+10|=2x+8$$

My way:
standard procedure:
1st: $$x+10=2x+8$$
$$x=2$$
OR
2nd:$$-(x+10)=2x+8$$
$$x=-6$$
the condition $$x=i$$ is met:
Insufficient.

The original explanation:
(2) $$|x+10|=2x+8$$. The left hand side (LHS) is an absolute value, which is always non-negative, hence RHS must also be non-negative: $$2x+8\geq0$$ giving us $$x\geq-4$$. Now, for this range $$x+10$$ is positive, hence$$|x+10|=x+10$$. So, $$|x+10|=2x+8$$ can be written as $$x+10=2x+8$$, solving for $$x$$ gives $$x=2$$.
Sufficient.

Noting that $$2x+8\geq0$$ excludes the negative value and leaves off only one value. But what the heck is the mechanics behind this problem that makes the the good old way of solving inequalities insufficiently precise here?

Merging similar topics.

The way you call "standard procedure" is not complete.

When expanding $$|x+10|=2x+8$$ you should consider x<-10 range and x>=-10 range:

$$x<-10$$ --> $$-(x+10)=2x+8$$ --> $$x=-6$$. Discard this solution since it's not in the range $$x<-10$$.
$$x\geq{-10}$$ --> $$x+10=2x+8$$ --> $$x=2$$.

Hope it's clear.
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Re: What is the value of integer x?  [#permalink]

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Bunuel wrote:
LM wrote:
What is the value of integer x?

(1) 2x^2+9<9x
(2) |x+10|=2x+8

What is the value of integer x?

(1) 2x^2+9<9x --> factor qudratics: $$(x-\frac{3}{2})(x-3)<0$$ --> roots are $$\frac{3}{2}$$ and 3 --> "<" sign indicates that the solution lies between the roots: $$1.5<x<3$$ --> since there only integer in this range is 2 then $$x=2$$. Sufficient.

(2) |x+10|=2x+8 --> LHS is an absolute value, which is always non negative, hence RHS must also be non-negative: $$2x+8\geq{0}$$ --> $$x\geq{-4}$$, for this range $$x+10$$ is positive hence $$|x+10|=x+10$$ --> $$x+10=2x+8$$ --> $$x=2$$. Sufficient.

Check this for more on solving inequalities like the one in the first statement:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.

this wud be realy easy question..

how did u change this in fraction..?2x2+9<9x.. (x-3/2) (x-3) ... ?? I can change integers..bt this fraction _________________
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Re: What is the value of integer x?  [#permalink]

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sanjoo wrote:
Bunuel wrote:
LM wrote:
What is the value of integer x?

(1) 2x^2+9<9x
(2) |x+10|=2x+8

What is the value of integer x?

(1) 2x^2+9<9x --> factor qudratics: $$(x-\frac{3}{2})(x-3)<0$$ --> roots are $$\frac{3}{2}$$ and 3 --> "<" sign indicates that the solution lies between the roots: $$1.5<x<3$$ --> since there only integer in this range is 2 then $$x=2$$. Sufficient.

(2) |x+10|=2x+8 --> LHS is an absolute value, which is always non negative, hence RHS must also be non-negative: $$2x+8\geq{0}$$ --> $$x\geq{-4}$$, for this range $$x+10$$ is positive hence $$|x+10|=x+10$$ --> $$x+10=2x+8$$ --> $$x=2$$. Sufficient.

Check this for more on solving inequalities like the one in the first statement:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

Hope it helps.

this wud be realy easy question..

how did u change this in fraction..?2x2+9<9x.. (x-3/2) (x-3) ... ?? I can change integers..bt this fraction If you cannot factor directly, then solve 2x^2-9x+9=0 to find the roots and factor that way.

Hope this helps.
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Re: What is the value of integer x ? I- 2x^2 + 9 < 9x II- |x+10  [#permalink]

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What is the value of integer x ?

I- 2x^2 + 9 < 9x
II- |x+10| = 2x+8

I took some time but solved it my way.

IMO D.

Stmt I. we can make equation $$2x^2 -9x + 9 < 0$$

solve for X = 1.5, 3 -- 2 is the only integer in between satisfies the relation.

Sufficient.

II. Square both side.
$$(|x+10|) ^2=(2x+8)^2$$

$$x^2 + 4x - 12 = 0$$

solve for X = 2, -6 -- substitute values back to original equation, only 2 satisfies the relation.

Sufficient.
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Re: What is the value of integer x? (M27-14)  [#permalink]

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Sad..I carelessly took X>0 instead of X>10.. and hence both the answers for statement II fitted in Nice question
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Re: What is the value of integer x? (M27-14)  [#permalink]

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Bunuel wrote:
LM wrote:
What is the value of integer x?

(1) 2x^2+9<9x
(2) |x+10|=2x+8

What is the value of integer x?

(1) 2x^2+9<9x --> factor qudratics: $$(x-\frac{3}{2})(x-3)<0$$ --> roots are $$\frac{3}{2}$$ and 3 --> "<" sign indicates that the solution lies between the roots: $$1.5<x<3$$ --> since there only integer in this range is 2 then $$x=2$$. Sufficient.

(2) |x+10|=2x+8 --> LHS is an absolute value, which is always non negative, hence RHS must also be non-negative: $$2x+8\geq{0}$$ --> $$x\geq{-4}$$, for this range $$x+10$$ is positive hence $$|x+10|=x+10$$ --> $$x+10=2x+8$$ --> $$x=2$$. Sufficient.

Hi Bunuel,

Statement1: When we factor 2x^2+9<9x
we get (2x-3) (x-3) < 0
ie x< 3/2 x< 1.5 or x< 3
So, how did we arrive at 1.5< x< 3

Could you please explain?

Thanks
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Re: What is the value of integer x? (M27-14)  [#permalink]

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swatjazz wrote:
Hi Bunuel,

Statement1: When we factor 2x^2+9<9x
we get (2x-3) (x-3) < 0
ie x< 3/2 x< 1.5 or x< 3
So, how did we arrive at 1.5< x< 3

Could you please explain?

Thanks

Check this for more on solving inequalities like the one in the first statement:
http://gmatclub.com/forum/x2-4x-94661.html#p731476
http://gmatclub.com/forum/inequalities-trick-91482.html
http://gmatclub.com/forum/everything-is ... me#p868863
http://gmatclub.com/forum/xy-plane-7149 ... ic#p841486

Hope it helps.
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Re: What is the value of integer x? (M27-14)  [#permalink]

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Bunuel wrote:
swatjazz wrote:
Hi Bunuel,

Statement1: When we factor 2x^2+9<9x
we get (2x-3) (x-3) < 0
ie x< 3/2 x< 1.5 or x< 3
So, how did we arrive at 1.5< x< 3

Could you please explain?

Thanks

Check this for more on solving inequalities like the one in the first statement:
http://gmatclub.com/forum/x2-4x-94661.html#p731476
http://gmatclub.com/forum/inequalities-trick-91482.html
http://gmatclub.com/forum/everything-is ... me#p868863
http://gmatclub.com/forum/xy-plane-7149 ... ic#p841486

Hope it helps.

Thanks so much once again.
A very important concept i learnt today. Re: What is the value of integer x? (M27-14)   [#permalink] 30 Aug 2019, 06:30
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