For all x > 1, which of the following is equivalent to the expression \frac{2x^2 - 5x - 3}… — SAT

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

The correct answer is

1

. Factoring the numerator of the rational expression gives

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

. Since

x > 1

, the denominator

2x + 1

is non-zero, allowing the expression to be simplified to

x - 3

. Subtracting

(x - 4)

and distributing the negative sign to both terms inside the parentheses yields

(x - 3) - (x - 4) = x - 3 - x + 4 = 1

Factor the quadratic expression in the numerator of the rational term.

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

This allows for the identification of common factors that can be simplified with the denominator.

Simplify the rational expression by canceling the common factor in the numerator and denominator.

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

Since

x > 1

, the term

2x + 1

is positive and non-zero, making the division valid.

Subtract the linear expression from the simplified rational expression, distributing the negative sign to both terms inside the parentheses.

(x - 3) - (x - 4) = x - 3 - x + 4 = 1

This performs the final subtraction and simplifies the expression to its equivalent constant form.

Simplifying rational expressions by factoring and performing operations on equivalent expressions

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