What is the positive real solution to the equation \frac{3}{x-2} - \frac{2}{x} = 1? — ACT

What is the positive real solution to the equation $\frac{3}{x-2} - \frac{2}{x} = 1$ ?

Answer: 4

The positive real solution to the equation is 4.

By multiplying the entire equation by the least common denominator,

x(x-2)

, the rational equation is cleared of fractions, resulting in

3x - 2(x-2) = x(x-2)

. Simplifying this yields the quadratic equation

x^2 - 3x - 4 = 0

. Factoring the quadratic equation gives

(x - 4)(x + 1) = 0

, which has roots of

x = 4

and

x = -1

. The positive solution is 4.

Multiply both sides of the equation by the least common denominator,

x(x-2)

, to eliminate the fractions.

3x - 2(x-2) = x(x-2)

Eliminating the denominators simplifies the rational equation into a polynomial equation.

Distribute and combine like terms to simplify both sides of the equation.

x + 4 = x^2 - 2x

Simplification is necessary before rearranging the equation into a solvable form.

Rearrange the terms into standard quadratic form:

ax^2 + bx + c = 0

x^2 - 3x - 4 = 0

Setting the quadratic expression equal to zero allows for factoring and solving.

Factor the quadratic equation.

(x - 4)(x + 1) = 0

Finding the factors reveals the potential values for

x

Solve for the roots and select the positive real solution.

x = 4

The equation yields two solutions,

x = 4

and

x = -1

, and the question explicitly requests the positive solution.

Solving rational equations by converting them to quadratic equations using the least common denominator.

Question