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# The operator @ is defined by the following expression: a@b = |(a + 1)|

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Math Expert
Joined: 02 Sep 2009
Posts: 51301
The operator @ is defined by the following expression: a@b = |(a + 1)|  [#permalink]

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27 Jun 2018, 20:14
00:00

Difficulty:

95% (hard)

Question Stats:

41% (03:06) correct 59% (03:00) wrong based on 131 sessions

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The operator $$@$$ is defined by the following expression: $$a@b = |\frac{a+1}{a}| - \frac{b+1}{b}$$ where $$ab\neq{0}$$. What is the sum of the solutions to the equation $$x@2 = \frac{x@(-1)}{2}$$ ?

A. -1

B. -0.75

C. -0.25

D. 0.25

E. 0.75

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Math Expert
Joined: 02 Aug 2009
Posts: 7113
The operator @ is defined by the following expression: a@b = |(a + 1)|  [#permalink]

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28 Jun 2018, 04:08
Bunuel wrote:
The operator $$@$$ is defined by the following expression: $$a@b = |\frac{a+1}{a}| - \frac{b+1}{b}$$ where $$ab\neq{0}$$. What is the sum of the solutions to the equation $$x@2 = \frac{x@(-1)}{2}$$ ?

A. -1

B. -0.75

C. -0.25

D. 0.25

E. 0.75

Before we go to actual calculations, let's look at what we have...
1) We have x on each side, with function of x on left side and $$\frac{(function of x)}{2}$$ on right side..
So final will be $$\frac{(function of x)}{2}$$on left side...
2) @-1 will leave 0 as -1+1 is zero

So $$\frac{1}{2}*(|\frac{x+1}{x}|)-|\frac{2+1}{2}|=0$$......
$$|\frac{x+1}{x}|=3$$....
Two cases..
1) $$\frac{(x+1)}{x}=3......x+1=3x....x=\frac{1}{2}$$
2) $$\frac{(x+1)}{x}=-3.....x+1=-3x.....x=-\frac{1}{4}$$
Sum = $$\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$$ or 0.25

even if you solve it completely
$$x@2 = |\frac{x+1}{x}| - \frac{2+1}{2}=|\frac{x+1}{x}| - \frac{3}{2}$$.....
$$\frac{x@-1}{2} = \frac{1}{2}*|\frac{x+1}{x}| - \frac{-1+1}{1}=\frac{1}{2}*|\frac{x+1}{x}|$$
so $$|\frac{x+1}{x}| - \frac{3}{2}=\frac{1}{2}*|\frac{x+1}{x}|........................\frac{1}{2}|\frac{x+1}{x}| = \frac{3}{2}...................|\frac{x+1}{x}| = 3$$
Two cases..
1) $$\frac{(x+1)}{x}=3......x+1=3x....x=\frac{1}{2}$$
2) $$\frac{(x+1)}{x}=-3.....x+1=-3x.....x=-\frac{1}{4}$$

D
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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The operator @ is defined by the following expression: a@b = |(a + 1)| &nbs [#permalink] 28 Jun 2018, 04:08
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