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

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

\(|\frac{x+1}{x}|=3\)

I didnt understand this question

|(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

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

So, Answer will be D
Hope it helps!

Watch the following video to learn the Basics of Functions and Custom Characters

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

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

Answer D