For all x > 1, which of the following is equivalent to the expression \frac{4x^2 - 1}{2x -… — SAT

For all $x > 1$ , which of the following is equivalent to the expression $\frac{4x^2 - 1}{2x - 1} - \frac{6x^2 + x - 2}{3x + 2}$ ?

Factoring the numerators allows the expression to be simplified. The first term becomes

2x + 1

, and the second term becomes

2x - 1

. Subtracting the second term from the first and distributing the negative sign results in

(2x + 1) - (2x - 1) = 2x + 1 - 2x + 1 = 2

Factor the numerator of the first rational term,

4x^2 - 1

, using the difference of squares identity.

4x^2 - 1 = (2x - 1)(2x + 1)

To identify and divide out common factors between the numerator and denominator.

Simplify the first term by dividing the factored numerator by its denominator,

2x - 1

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

Since

x > 1

2x - 1 \neq 0

, allowing the division.

Factor the numerator of the second rational term,

6x^2 + x - 2

6x^2 + x - 2 = (2x - 1)(3x + 2)

To find common factors that can be simplified with the denominator.

Simplify the second term by dividing the factored numerator by its denominator,

3x + 2

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

Since

x > 1

3x + 2 \neq 0

, allowing the division.

Subtract the second simplified expression from the first, ensuring that the negative sign is correctly distributed to all terms.

(2x + 1) - (2x - 1) = 2x + 1 - 2x + 1 = 2

To combine the terms and find the final equivalent value of the entire expression.

Equivalent Algebraic Expressions

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