For all x > 1, the expression \frac{2x^2 + 7x - 4}{x^2 - 1} \cdot \frac{x - 1}{2x - 1} is … — SAT

For all $x > 1$ , the expression $\frac{2x^2 + 7x - 4}{x^2 - 1} \cdot \frac{x - 1}{2x - 1}$ is equivalent to $\frac{x+k}{x+1}$ , where $k$ is a constant. What is the value of $k$ ?

Answer: 4

Answer

The value of the constant

k

is 4.

Factoring the numerator

2x^2 + 7x - 4

yields

(2x - 1)(x + 4)

and factoring the denominator

x^2 - 1

yields

(x - 1)(x + 1)

. Substituting these factored forms into the given product gives

\frac{(2x - 1)(x + 4)}{(x - 1)(x + 1)} \cdot \frac{x - 1}{2x - 1}

. Canceling the common factors

(2x - 1)

and

(x - 1)

simplifies the expression to

\frac{x + 4}{x + 1}

. Comparing this to

\frac{x + k}{x + 1}

shows that

k = 4

Step-by-Step Solution

Factor the quadratic expression in the numerator:

2x^2 + 7x - 4

(2x - 1)(x + 4)

Factoring the numerator helps identify common factors that can be simplified.

Factor the difference of squares in the denominator:

x^2 - 1

(x - 1)(x + 1)

Factoring the denominator helps identify common factors that can be simplified.

Multiply the rational expressions and cancel out the common factors.

\frac{x + 4}{x + 1}

Since

x > 1

, the terms

(2x - 1)

and

(x - 1)

are not equal to zero and can be canceled.

Compare the resulting expression with

\frac{x + k}{x + 1}

to find the value of

k

k = 4

By matching the numerators of the equivalent expressions,

x + 4 = x + k

, which gives

k = 4

Key Concept

Factoring and simplifying products of rational expressions

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