1954 AMC 12 Problem 22

Question

The expression

\dfrac{2x^2-x}{(x+1)(x-2)}

{}-\dfrac{4+x}{(x+1)(x-2)}

cannot be evaluated for

x=-1

or

x=2,

since division by zero is not allowed. For other values of

x:

The expression takes on many different values.

The expression has only the value

2.

The expression has only the value

1.

The expression always has a value between

-1

and

2.

The expression has a value greater than

2

or less than

-1.

Combine the two fractions over their common denominator

The numerator factors as

2(x+1)(x-2)

Combining the numerators gives

\begin{gathered} 2x^2-x-(4+x)\\ =2x^2-2x-4\\ =2(x+1)(x-2). \end{gathered}

For the allowed values

x\ne-1,2,

the factors cancel and the expression equals

2.

Thus, the correct answer is B.

Problem 22 in other years