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

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Re: The operator @ is defined by the following expression: a@b = |(a + 1)| [#permalink]
So $$\frac{1}{2}*(|\frac{x+1}{x}|)-|\frac{2+1}{2}|=0$$......
$$|\frac{x+1}{x}|=3$$....

Can anyone explain how $$|\frac{x+1}{x}|=3$$?
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Re: The operator @ is defined by the following expression: a@b = |(a + 1)| [#permalink]
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harshit121 wrote:
So $$\frac{1}{2}*(|\frac{x+1}{x}|)-|\frac{2+1}{2}|=0$$......
$$|\frac{x+1}{x}|=3$$....

Can anyone explain how $$|\frac{x+1}{x}|=3$$?

$$\frac{1}{2}*(|\frac{x+1}{x}|)-|\frac{2+1}{2}|=0$$......

=> $$\frac{1}{2}*(|\frac{x+1}{x}|) = \frac{3}{2}$$

Multiply L.H.S. and R.H.S. by 2

$$|\frac{x+1}{x}|=3$$
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Re: The operator @ is defined by the following expression: a@b = |(a + 1)| [#permalink]
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I didnt understand this question
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Re: The operator @ is defined by the following expression: a@b = |(a + 1)| [#permalink]
|(X+1)/X|=3

If (X+1)/X < 0, then X<-1
On that basis I started solving
(X+1)/X = -3
X + 1 = -3X
X = -1/4 or -0.25

this solution did not match the criteria as -0.25 is not less than -1
So I rejected it. Where am I going wrong? Bunuel
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Re: The operator @ is defined by the following expression: a@b = |(a + 1)| [#permalink]
smw wrote:
|(X+1)/X|=3

If (X+1)/X < 0, then X<-1
On that basis I started solving
(X+1)/X = -3
X + 1 = -3X
X = -1/4 or -0.25

this solution did not match the criteria as -0.25 is not less than -1
So I rejected it. Where am I going wrong? Bunuel

If $$|\frac{x+1}{x}|=3$$

$$|\frac{x+1}{x}|<0$$

So, both x+1 and x are of opposite sign.

Two cases
1) x>0, then x+1<0 or x<-1…….NO
2) x<0, then x+1>0 or x>-1…..YES
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Re: The operator @ is defined by the following expression: a@b = |(a + 1)| [#permalink]
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Given that $$a@b = |\frac{a+1}{a}| - \frac{b+1}{b}$$ where $$ab\neq{0}$$ and we need to find the sum of all solutions of the equation $$x@2 = \frac{x@(-1)}{2}$$

Let's start by finding the value of x@2 first
To find the value of x@2, compare x@2 with a@b
=> a=x and b=2
So, to find x@2 substitute a=x and b=2 in $$a@b = |\frac{a+1}{a}| - \frac{b+1}{b}$$
=> $$x@2 = |\frac{x+1}{x}| - \frac{2+1}{2}$$ =$$|\frac{x+1}{x}| - \frac{3}{2}$$

Similarly, $$x@(-1) = |\frac{x+1}{x}| - \frac{-1+1}{-1}$$ =$$| \frac{x+1}{x}| - 0$$ = $$|\frac{x+1}{x}|$$

Given that $$x@2 = \frac{x@(-1)}{2}$$
=> $$|\frac{x+1}{x}| - \frac{3}{2}$$ = $$|\frac{x+1}{x}|$$ / 2
=> $$| \frac{x+1}{x}| - |\frac{x+1}{x}|$$ / 2 =$$\frac{3}{2}$$
Multiply both the sides by 2 we get
$$2*|\frac{x+1}{x}| - |\frac{x+1}{x}|$$ = 3
=> $$|\frac{x+1}{x}|$$ = 3

We will get two cases (Watch this video to know about the basic of Absolute Value)
Case 1: $$\frac{x+1}{x }$$>= 0 (0r x+1 > 0 => x > -1 and x≠0
=> $$|\frac{x+1}{x}|$$ = $$\frac{x+1}{x }$$
=> $$\frac{x+1}{x }$$ = 3
=> x + 1 = 3x
=> 2x = 1 => x = $$\frac{1}{2}$$ which is > -1 so this is one solution

Case 2: $$\frac{x+1}{x }$$< 0
=> $$|\frac{x+1}{x}|$$ = - $$\frac{x+1}{x }$$
=> -$$\frac{x+1}{x }$$ = 3
=> -x - 1 = 3x => 4x = -1
or x = $$\frac{-1}{4}$$
Let's check if this is a right answer by substituting the value in $$\frac{x+1}{x }$$< 0
((-1/4) + 1) / (-1/4) = (3/4) / (-1/4) < 0 so this is also a solution

=> Sum of values of the solution = $$\frac{1}{2}$$ + $$\frac{-1}{4}$$ = $$\frac{1}{4}$$ = 0.25

Hope it helps!

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Re: The operator @ is defined by the following expression: a@b = |(a + 1)| [#permalink]
Hi BrushMyQuant,

Thank you for the solution! Can you kindly elaborate more on the steps before Case 1 and 2? Why was this necessary? $$|\frac{x+1}{x}|$$ = 3

Many thanks,
Em
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Re: The operator @ is defined by the following expression: a@b = |(a + 1)| [#permalink]
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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

a@b = |(a+1)/a| - (b+1)/b
We need to solve: x@2 = 1/2 * [x@(-1)]
i.e. 2 * x@2 = x@(-1)

We have: 2 * x@2 = 2 * [|(x+1)/x| - (2+1)/2] = 2 * |(x+1)/x| - 3
Also: x@(-1) = |(x+1)/x| - (-1+1)/2 = |(x+1)/x| - 0

Thus, we have: 2 * |(x+1)/x| - 3 = |(x+1)/x|
=> |(x+1)/x| = 3
=> (x+1)/x = 3 or -3
=> x+1 = 3x or -3x
=> x = 1/2 or -1/4

Sum of the solutions = 1/2 + (-1/4) = 1/4