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Absolute & Inequality tricky question...Expert needed

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by Mo2men » Sun Nov 27, 2016 2:12 am
If |x| < 20 and |x - 8| > |x + 4|, which of the following expresses the allowable range for x?

(A) -12 < x < 12

(B) -20 < x < 2

(C) -20 < x < -12 and 12 < x < 20

(D) -20 < x < -8 and 4 < x < 20

(E) -20 < x < -4 and 8 < x < 20

Source: Magoosh
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by GMATGuruNY » Sun Nov 27, 2016 4:25 am
Mo2men wrote:If |x| < 20 and |x - 8| > |x + 4|, which of the following expresses the allowable range for x?

(A) -12 < x < 12

(B) -20 < x < 2

(C) -20 < x < -12 and 12 < x < 20

(D) -20 < x < -8 and 4 < x < 20

(E) -20 < x < -4 and 8 < x < 20
Test x=10 in |x - 8| > |x + 4|:
|10 - 8| > |10 + 4|
2 > 14
Does not work.
Eliminate any answer choice with a range that includes x=10.
Eliminate A, D and E.

Test x=0 in |x - 8| > |x + 4|:
|0 - 8| > |0 + 4|
8 > 4
This works.
Eliminate any remaining answer choice that does not allow x=0.
Eliminate C.

The correct answer is B.
Last edited by GMATGuruNY on Sun Nov 27, 2016 4:37 am, edited 2 times in total.
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by Mo2men » Sun Nov 27, 2016 4:35 am
GMATGuruNY wrote:
Mo2men wrote:If |x| < 20 and |x - 8| > |x + 4|, which of the following expresses the allowable range for x?

(A) -12 < x < 12

(B) -20 < x < 2

(C) -20 < x < -12 and 12 < x < 20

(D) -20 < x < -8 and 4 < x < 20

(E) -20 < x < -4 and 8 < x < 20
Test x=10 in |x - 8| > |x + 4|:
|10 - 8| > |10 + 4|
2 > 14
Does not work.
Eliminate any answer choice with a range that includes x=10.
Eliminate A, D and E.

Test x=0 in |x - 8| > |x + 4|:
|0 - 8| > |0 + 4|
8 > 4
This works.
Eliminate any answer choice that does not allow x=10.
Eliminate C.

The correct answer is B.
Thanks Mitch but i think you mean x=0 in the highlighted part above.
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by GMATGuruNY » Sun Nov 27, 2016 4:39 am
Mo2men wrote:
[Eliminate any answer choice that does not allow x=10.
Eliminate C.

The correct answer is B.
Thanks Mitch but i think you mean x=0 in the highlighted part above.
Good catch.
The typo has been corrected.
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by GMATGuruNY » Sun Nov 27, 2016 2:49 pm
Mo2men wrote:If |x| < 20 and |x - 8| > |x + 4|, which of the following expresses the allowable range for x?

(A) -12 < x < 12

(B) -20 < x < 2

(C) -20 < x < -12 and 12 < x < 20

(D) -20 < x < -8 and 4 < x < 20

(E) -20 < x < -4 and 8 < x < 20

Source: Magoosh
Alternate approach:
When both sides of an inequality are enclosed in absolute value symbols, we can SQUARE THE INEQUALITY.

|x - 8|² > |x + 4|²
x² + 64 - 16x > x² + 16 + 8x
48 > 24x
2 > x
x < 2.
Since |x|<20 constrains x to values between -20 and 20, -20 < x < 2.

The correct answer is B.
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by GMATGuruNY » Sun Nov 27, 2016 3:18 pm
Mo2men wrote:If |x| < 20 and |x - 8| > |x + 4|, which of the following expresses the allowable range for x?

(A) -12 < x < 12

(B) -20 < x < 2

(C) -20 < x < -12 and 12 < x < 20

(D) -20 < x < -8 and 4 < x < 20

(E) -20 < x < -4 and 8 < x < 20

Source: Magoosh
Alternate approach 2:
|a-b| = the distance between a and b.
|a+b| = |a-(-b)| = the distance between a and -b.

|x - 8| > |x + 4|.
In words:
The distance between x and 8 is greater than the distance between x and -4.
In other words, x is CLOSER TO -4 THAN TO 8.
Plotted on a number line:
<----- (-4) ---- 2 ---- 8 ----->
Since 2 is halfway between -4 and 8, the blue portion is composed of all values closer to -4 than to 8.
The blue portion indicates that x<2.
|x|<20 constrains x to values between -20 and 20.
Thus:
-20 < x < 2.

The correct answer is B.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
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I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
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