For all x > 2, which of the following is equivalent to the expression \frac{3x^2 - 12}{x^2… — SAT

For all $x > 2$ , which of the following is equivalent to the expression $\frac{3x^2 - 12}{x^2 - 2x} - \frac{2x^2 + 2x - 12}{x^2 + 3x}$ ?

\frac{x+10}{x}

The correct answer is obtained by factoring both rational expressions and simplifying them before subtracting. The first term factors into

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

, which simplifies to

\frac{3(x+2)}{x}

. The second term factors into

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

, which simplifies to

\frac{2(x-2)}{x}

. Subtracting these two expressions yields

\frac{3(x+2) - 2(x-2)}{x} = \frac{3x+6-2x+4}{x} = \frac{x+10}{x}

Factor the numerators and denominators of both rational expressions.

The expression becomes

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

Factoring helps identify common binomial factors in the numerator and denominator.

Simplify both terms by canceling common factors.

For

x > 2

, the expression simplifies to

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

Since

x > 2

, the terms

x-2

and

x+3

are non-zero and can be canceled.

Combine the simplified terms over the common denominator

x

\frac{3(x+2) - 2(x-2)}{x} = \frac{3x + 6 - (2x - 4)}{x}

Both terms share the common denominator

x

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

\frac{3x + 6 - 2x + 4}{x} = \frac{x+10}{x}

Distributing subtraction to the terms inside

(2x-4)

yields

-2x + 4

Simplifying and subtracting rational expressions by factoring and finding a common denominator

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