What is the value of the larger real solution to the equation \frac{10}{x} - \frac{3}{x-1}… — ACT

What is the value of the larger real solution to the equation $\frac{10}{x} - \frac{3}{x-1} = 2$ ?

Answer: 2.5

The larger real solution is

2.5

Multiplying the equation by the least common denominator

x(x-1)

yields

10(x-1) - 3x = 2x(x-1)

. Simplifying this leads to

2x^2 - 9x + 10 = 0

. Factoring the quadratic yields

(2x-5)(x-2) = 0

, which gives the solutions

x = 2.5

and

x = 2

. The larger of these two solutions is

2.5

Multiply the entire equation by the least common denominator, which is

x(x-1)

, for

x \neq 0

and

x \neq 1

10(x-1) - 3x = 2x(x-1)

This clears the fractions from the rational equation.

Expand both sides of the equation.

10x - 10 - 3x = 2x^2 - 2x

Expanding the terms allows us to combine like terms.

Combine like terms on the left side and move all terms to one side to set the quadratic equation equal to zero.

2x^2 - 9x + 10 = 0

Setting the quadratic equation to zero is required to solve it by factoring.

Factor the quadratic equation by grouping.

(2x-5)(x-2) = 0

Factoring allows us to find the roots of the quadratic equation.

Solve for

x

by setting each factor equal to zero.

x = 2.5

x = 2

By the zero product property, at least one of the factors must be zero.

Check for extraneous solutions and select the larger real value.

Both

2

and

2.5

are valid because they do not make the original denominators zero. The larger value is

2.5

The question specifically asks for the larger of the two real solutions.

Solving rational equations by clearing denominators

Estimated Time:1m 30s

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