For all x > 2, which of the following is equivalent to the expression \frac{3x^2 - 5x - 2}… — SAT

For all $x > 2$ , which of the following is equivalent to the expression $\frac{3x^2 - 5x - 2}{x^2 - 4} - \frac{x - 1}{x + 2}$ ?

The expression is equivalent to

\frac{2x + 2}{x + 2}

The correct answer is obtained by first factoring the first term:

\frac{3x^2 - 5x - 2}{x^2 - 4} = \frac{(3x + 1)(x - 2)}{(x - 2)(x + 2)}

. Canceling the common factor of

x - 2

yields

\frac{3x + 1}{x + 2}

. Subtracting the second term gives

\frac{3x + 1 - (x - 1)}{x + 2} = \frac{3x + 1 - x + 1}{x + 2} = \frac{2x + 2}{x + 2}

Factor the numerator and the denominator of the first term of the expression.

The numerator factors as

3x^2 - 5x - 2 = (3x + 1)(x - 2)

, and the denominator factors as

x^2 - 4 = (x - 2)(x + 2)

This allows common factors in the numerator and denominator to be identified and canceled.

Simplify the first term by canceling the common factor

(x - 2)

for

x > 2

The first term simplifies to

\frac{3x + 1}{x + 2}

For

x > 2

x - 2 \neq 0

, so we can divide both numerator and denominator by

x - 2

to simplify the fraction.

Subtract the second term from the simplified first term.

The expression becomes

\frac{3x + 1}{x + 2} - \frac{x - 1}{x + 2} = \frac{(3x + 1) - (x - 1)}{x + 2}

Since both fractions have the same denominator,

x + 2

, their numerators can be subtracted directly.

Distribute the negative sign in the numerator and combine like terms.

\frac{3x + 1 - x + 1}{x + 2} = \frac{2x + 2}{x + 2}

Distributing the subtraction to both terms in the parenthesis

(x - 1)

gives

-x + 1

. Combining

3x - x

yields

2x

, and

1 + 1

yields

2

Simplifying rational expressions by factoring and performing algebraic operations with common denominators.

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