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D
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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
A. -1
B. -0.75
C. -0.25
D. 0.25
E. 0.75
28 Jun 2018, 05:08
Kudos
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Bunuel
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
16 Dec 2021, 06:40
Kudos
Bookmarks
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
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
