What is the larger solution to the equation below? \frac{x}{x - 2} + \frac{3}{x} = 2 — SAT

What is the larger solution to the equation below?

\frac{x}{x - 2} + \frac{3}{x} = 2

Answer: 6

The larger solution to the equation is 6.

To solve the rational equation, we clear the denominators by multiplying both sides by

x(x - 2)

, which yields the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation gives

(x - 6)(x - 1) = 0

, leading to the solutions

x = 6

and

x = 1

. Both values are valid because they do not make any denominator of the original expression equal to zero. The larger of the two values is 6.

Multiply both sides of the equation by the common denominator

x(x - 2)

to clear the fractions.

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

Clearing denominators simplifies the rational equation into a polynomial equation.

Expand both sides of the equation.

x^2 + 3x - 6 = 2x^2 - 4x

Distributing the multiplication allows us to combine like terms.

Move all terms to one side of the equation to write it in the standard quadratic form

ax^2 + bx + c = 0

x^2 - 7x + 6 = 0

Setting the quadratic expression to zero prepares it for factoring.

Factor the quadratic equation.

(x - 6)(x - 1) = 0

Finding two numbers that multiply to

6

and sum to

-7

gives

-6

and

-1

Identify the values of

x

that satisfy the factored equation and check for extraneous solutions.

x = 6

x = 1

Neither solution makes the denominators in the original equation,

x-2

x

, equal to zero, so both are valid. The larger of these two solutions is 6.

Solving rational equations by clearing denominators to form a quadratic equation, and verifying solutions against the domain constraints.

Question