If x is a real number that satisfies the equation below, what is the value of x? \frac{x +… — SAT

x

is a real number that satisfies the equation below, what is the value of

x

\frac{x + 2}{x - 4} - \frac{3}{x} = \frac{12}{x^2 - 4x}

Answer: 1

The only valid real solution to the equation is

1

Multiplying the equation by the common denominator

x(x - 4)

simplifies the equation to

x^2 - x = 0

. Solving this quadratic equation yields

x = 0

and

x = 1

. Because

x = 0

leads to a division by zero in the original equation, it is extraneous. Therefore, the only valid real solution is

1

Determine the common denominator and restrictions for the rational equation.

The common denominator is

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

. The restrictions are

x \neq 0

and

x \neq 4

Finding the common denominator allows us to eliminate fractions, while the restrictions help us identify potential extraneous solutions.

Multiply the entire equation by the common denominator

x(x - 4)

x(x + 2) - 3(x - 4) = 12

This step clears the rational expressions, leaving a polynomial equation.

Expand and simplify the polynomial equation.

x^2 + 2x - 3x + 12 = 12

, which simplifies to

x^2 - x = 0

Expanding the terms allows us to combine like terms and set the quadratic equation to zero.

Factor the quadratic equation to solve for

x

x(x - 1) = 0

, giving candidate solutions

x = 0

x = 1

Applying the zero-product property identifies the roots of the quadratic equation.

Verify the candidate solutions against the initial restrictions.

Since

x = 0

makes the denominators in the original equation equal to zero, it is extraneous. The candidate solution

x = 1

is valid.

We must verify solutions because multiplying by a variable expression can introduce extraneous roots that make the original expressions undefined.

Solving rational equations by clearing denominators and checking for extraneous solutions.

Estimated Time:1m 30s

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