`x/(x-3)=(2x)/2`
Which of the following represents all the possible values of x that satisfy the equation above?

Options

A. 0 and 2
B. 0 and 4
C. `–4` and 4
D. 4

Answer

B

Explanation

The right-hand side of the given equation, `(2x)/2` , can be rewritten as x. Multiplying both sides of the equation `x/(x-3)=x` by x − 3 yields x = x(x − 3). Applying the distributive property of multiplication to the right-hand side of the equation x = x(x − 3) yields `x = x^2 − 3x`. Subtracting x from both sides of this equation yields `0 = x^2 − 4x`. Factoring x from both terms of `x^2 − 4x` yields 0 = x(x − 4). By the zero product property, the solutions to the equation 0 = x(x − 4) are x = 0 and x − 4 = 0, or x = 4. Substituting 0 and 4 for x in the given equation yields 0 = 0 and 4 = 4, respectively. Since both are true statements, both 0 and 4 are solutions to the given equation.
Choice A is incorrect and may result from a sign error. Choice C is incorrect and may result from an error in factoring. Choice D is incorrect and may result from not considering 0 as a possible solution.