M41-07

Question

If  x ≠ 1  and  \frac{|x|}{16} + \frac{x}{8} + \frac{|x|}{4} + \frac{x}{2} = -1 , what is the value of  \frac{-|x| - 1}{x - 1} ?

A.  1 
B.  \frac{1}{2} 
C.  0 
D.  -\frac{1}{2} 
E.  -1

Bunuel · Answer

Official Solution:

If  x ≠ 1  and  \frac{|x|}{16} + \frac{x}{8} + \frac{|x|}{4} + \frac{x}{2} = -1 , what is the value of  \frac{-|x| - 1}{x - 1} ?

A.  1 
B.  \frac{1}{2} 
C.  0 
D.  -\frac{1}{2} 
E.  -1

Notice that if  x  is positive, then  \frac{|x|}{16} + \frac{x}{8} + \frac{|x|}{4} + \frac{x}{2} = 	ext{positive} + 	ext{positive} + 	ext{positive} + 	ext{positive} = 	ext{positive} , not a negative number  -1 . So,  x  cannot be positive.
 
When  x \leq 0 , then  |x| = -x , and in this case  \frac{-|x| - 1}{x - 1} = \frac{-(-x) - 1}{x - 1} = \frac{x - 1}{x - 1} = 1 .

Answer: A

divyasaxena2825 · Answer

I did not understand this. If modulus sign is removed , the digit will be positive.
For example , -|-5| = -5 right ? Please guide

Bunuel · Answer

Yes, -|-5| = -5. However, I’m not sure I fully understand your question. Could you please elaborate on what you mean? Thank you!

TanyaBhargava · Answer

I think she means why we did -|-x| = x. 
I know its because when x is -ve , |x| = -x but even i dont get it, when do we have to ignore mod (take negative sign) ? 
i guess when x is negative in that range mod opens to be negative?

TanyaBhargava · Answer

I tried solving for x, got x = -16/5 then put that in the mod, got 1. i dont know if this is conceptually right. can someone confirm?

Bunuel · Answer

First of all, |x| = |-x| for all values of x because |x| represents the distance of x from 0, and the distance of -x from 0 is the same.

Next:
 
 |x| = x when x ≥ 0 (e.g., |5| = 5). 
 |x| = -x when x ≤ 0 (e.g., |-5| = -(-5) = 5).

Finally, where in the solution do we have -|-x| = x? We have -|x|, which given that x ≤ 0 transforms to -(-x) = x.

Bunuel · Answer

Yes,  \frac{|x|}{16} + \frac{x}{8} + \frac{|x|}{4} + \frac{x}{2} = -1  gives x = -16/5. However, you do not need to solve this equation. As shown in the solution, you need to deduce that x is negative, and for ALL negative values of x,  \frac{-|x| - 1}{x - 1} = \frac{-(-x) - 1}{x - 1} = \frac{x - 1}{x - 1} = 1 .

kavyavijay008 · Answer

Can you please explain how did |x| is equal to -x?

Bunuel · Answer

|x| = x when x ≥ 0 (e.g., |5| = 5). 
 |x| = -x when x ≤ 0

Since we deduced that x must be less than 0, then |x| = -x.

10. Absolute Value 
 
    Theory    
 
 Math: Absolute value (Modulus) 
 Absolute Value: Tips and hints 
 Properties of Absolute Values on the GMAT 
 Absolute modulus : A better understanding 
 GMAT Question Patterns: Abolute Value 
 The Reason Behind Absolute Value Questions on the GMAT 
 
    Questions    
 
 Inequality and absolute value questions from my collection 
 The E-GMAT Question Series on ABSOLUTE VALUE 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

giuseppemvn · Answer

I like the solution - it’s helpful.

maanyack · Answer

So I found a simpler solution
first take 16 as the common denominator in the given equation:
|x|/16+x/8+|x|/4+x/2=−1 => 5|x| + 10x=-16

Now there are two possibilities either |x| is -x or x
case 1
|x|= -x
we get x=-16/5

case 2 
|x| = x
we get x= -16/15 which is invalid since x has to be positive in this case

now putting the case 1 value of x in the second equation of the question −|x|−1/x−1

x= -16/15, you'll get -16-15/-16-15 = 1
Ans: A

anushree01 · Answer

I like the solution - it’s helpful.

M41-07

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards