What is the value of integer x? (M27-14)

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.

Answer: D.

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
xy-plane-71492.html?hilit=solving%20quadratic#p841486

Hope it helps.
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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

Archit

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

thnks that was subtle i frgot to check the range.....

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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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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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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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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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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If you cannot factor directly, then solve 2x^2-9x+9=0 to find the roots and factor that way.

Factoring Quadratics: https://www.purplemath.com/modules/factquad.htm

Solving Quadratic Equations: https://www.purplemath.com/modules/solvquad.htm

Hope this helps.
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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.

Sad..I carelessly took X>0 instead of X>10.. and hence both the answers for statement II fitted in
Nice question

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.

Answer: D.

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:
https://gmatclub.com/forum/x2-4x-94661.html#p731476
https://gmatclub.com/forum/inequalities-trick-91482.html
https://gmatclub.com/forum/everything-is ... me#p868863
https://gmatclub.com/forum/xy-plane-7149 ... ic#p841486

Hope it helps.
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Thanks so much once again.
A very important concept i learnt today.

What is the value of integer x?

(1) 2x^2+9<9x --> factor qudratics: (x−32)(x−3)<0(x−32)(x−3)<0 --> roots are 3232 and 3 --> "<" sign indicates that the solution lies between the roots: 1.5<x<31.5<x<3 --> since there only integer in this range is 2 then x=2x=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≥02x+8≥0 --> x≥−4x≥−4, for this range x+10x+10 is positive hence |x+10|=x+10|x+10|=x+10 --> x+10=2x+8x+10=2x+8 --> x=2x=2. Sufficient.


I understand how you arrive at (x - 3/2)(x-3) < 0 and that one is positive and one is negative. But how do you determine that 3/2<x<3 as opposed to x<3/2 and x>3 ?

is it just me or do I only see 2 options? how can the answer be D if i can't see the option for D?

Hi, and welcome to GMAT Club.

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following post ALL YOU NEED FOR QUANT.

Hope this helps.
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