Solve 9 / x-5 = 5 / 2x-5

Question

Solve the equation

x_{1} = - \frac{3 10}{5}, x_{2} = \frac{3 10}{5}

Alternative Form

x_{1} \approx - 1.897367, x_{2} \approx 1.897367

Evaluate

\frac{9}{x} - 5 = \frac{5}{2} x - 5

Find the domain

\frac{9}{x} - 5 = \frac{5}{2} x - 5, x \neq = 0

Simplify

\frac{9}{x} = \frac{5}{2} x

Rewrite the expression

\frac{9}{x} = \frac{5 x}{2}

Cross multiply

9 \times 2 = x \times 5 x

Simplify the equation

18 = x \times 5 x

Simplify the equation

18 = 5 x^{2}

Swap the sides of the equation

5 x^{2} = 18

Divide both sides

\frac{5 x ^{2}}{5} = \frac{18}{5}

Divide the numbers

x^{2} = \frac{18}{5}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm \frac{18}{5}

Simplify the expression

More Steps

Evaluate

\frac{18}{5}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{18}{5}

Simplify the radical expression

More Steps

Evaluate

18

Write the expression as a product where the root of one of the factors can be evaluated

9 \times 2

Write the number in exponential form with the base of 3

3^{2} \times 2

The root of a product is equal to the product of the roots of each factor

3^{2} \times 2

Reduce the index of the radical and exponent with 2

32

\frac{3 2}{5}

Multiply by the Conjugate

\frac{3 2 \times 5}{5 \times 5}

Multiply the numbers

More Steps

Evaluate

2 \times 5

The product of roots with the same index is equal to the root of the product

2 \times 5

Calculate the product

10

\frac{3 10}{5 \times 5}

When a square root of an expression is multiplied by itself,the result is that expression

\frac{3 10}{5}

x = \pm \frac{3 10}{5}

Separate the equation into 2 possible cases

x = \frac{3 10}{5} x = - \frac{3 10}{5}

Check if the solution is in the defined range

x = \frac{3 10}{5} x = - \frac{3 10}{5}, x \neq = 0

Find the intersection of the solution and the defined range

x = \frac{3 10}{5} x = - \frac{3 10}{5}

Solution

x_{1} = - \frac{3 10}{5}, x_{2} = \frac{3 10}{5}

Alternative Form

x_{1} \approx - 1.897367, x_{2} \approx 1.897367

Show Solution