`(2)/(x+1)-(x)/(6)=0`
If `x_(1) and x_(2)` are valid solutions to the rational equation given, what is the sum of `x_(1)+x_(2)`?

Options

A. `-1`
B. `0`
C. `1`
D. `2`

Answer

A

Explanation

Difficult: Medium
Category: Passport to Advanced Math/ Exponents
Strategic advice: When solving a rational equation (an equation with variable in the denominator ) with exactly two terms, it is almost always quicker to set one term equal to the other (separate the terms on opposite sides of the=sign) and cross-multiply. You could also find the a common denominator and go grom there, but the usually takes more time.
Getting to the Answer: Add `(x)/(6)` to both sides of the equation,and then cross-multiply:
`(2)/(x+1)-(x)/(6)=0`
`(2)/(x+2)=(x)/(6)`
`x(x+1)=6(2)`
`x^(2)+x=12`
You now have a quadratic equation to solve. Start by subracting 12 from both sides so that the equation equal to 0. The answer choices aare all integer, so this is a clue that you should be able to factor the equation:
`x^(2)+x=12`
`(x+4)(x-3)=0`
Set each factor equal to 0 and solve to find that `x=-4 and x=3`. The sum of these values is `-4+3=-1`, so (A) is correct.