sat-dog

Choice C is correct. Each fraction in the given equation can be expressed with the common denominator . Multiplying by yields , and multiplying by yields . Therefore, the given equation can be written as . Multiplying each fraction by the denominator results in the equation , or . This equation can be solved by setting a quadratic expression equal to 0, then solving for x. Subtracting from both sides of this equation yields . The expression can be factored, resulting in the equation . By the zero product property, or . To solve for x in , 1 can be added to both sides of the equation, resulting in . Dividing both sides of this equation by 2 results in . Solving for x in yields . However, this value of x would result in the second fraction of the original equation having a denominator of 0. Therefore, is an extraneous solution. Thus, the only value of x that satisfies the given equation is .

Choice A is incorrect and may result from solving but not realizing that this solution is extraneous because it would result in a denominator of 0 in the second fraction. Choice B is incorrect and may result from a sign error when solving for x. Choice D is incorrect and may result from a calculation error.