The operator @ is defined by the following expression: a@b = | a+1a| - b+1b where ab\neq0. What is the sum of the solutions to the equation x@2 = x@(-1)2 ?

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Fractions/Ratios/Decimals

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 $x@2 = \frac{x@(-1)}{2}$ ?

Answer Choices

Correct answer marked below

-1

-0.75

-0.25

0.25

Correct

0.75

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Explanation

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Official Explanation

Official

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}

\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…

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What this question tests

What it tests

Your ability to convert between fractions, ratios, and decimals and to reason proportionally.

Common trap

Cross-multiplying incorrectly or mixing up the ratio parts — a part:part ratio is not a fraction of the whole.

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