An equation is shown below. \frac{x}{x - 1} - \frac{2}{x + 2} = \frac{6}{x^2 + x - 2} What… — SAT

An equation is shown below.

\frac{x}{x - 1} - \frac{2}{x + 2} = \frac{6}{x^2 + x - 2}

What is the value of the real solution to the equation?

Answer: 2

The correct answer is 2.

The correct answer is 2. Multiplying both sides by the least common denominator

(x - 1)(x + 2)

results in the quadratic equation

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

. Simplifying this equation yields

x^2 + 2 = 6

, which has solutions

x = 2

and

x = -2

. However, substituting

x = -2

into the original equation results in division by zero, making it an extraneous solution. Therefore,

2

is the only valid real solution.

Multiply the entire equation by the least common denominator,

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

, to clear the denominators.

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

This simplifies the rational equation into a polynomial equation.

Expand the terms and simplify the equation.

x^2 + 2x - 2x + 2 = 6

, which simplifies to

x^2 + 2 = 6

Distributing the terms allows us to group like terms and solve for the variable.

Solve the quadratic equation for

x

x^2 = 4

, which gives

x = 2

x = -2

Subtracting 2 from both sides isolates the squared variable.

Check for extraneous solutions by substituting the potential solutions back into the original denominators.

For

x = -2

, the denominator

x + 2

becomes 0, which is undefined. For

x = 2

, all denominators are non-zero.

Solutions that make any denominator in the original equation equal to zero are extraneous and must be excluded.

Solving rational equations by finding a common denominator and checking for extraneous solutions.

Question