What is the answer to x/(1-1/x)=x+3 ?

The answer to x/(1-1/x)=x+3 is x= 3/2

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x/(1-1/x)=x+3

\frac{x}{1 - \frac{1}{x}} = x + 3

x = \frac{3}{2}

Decimal Notation

x = 1.5

Solution steps

\frac{x}{1 - \frac{1}{x}} = x + 3

Simplify

\frac{x}{1 - \frac{1}{x}} : \frac{x ^{2}}{x - 1}

\frac{x ^{2}}{x - 1} = x + 3

Multiply both sides by

x - 1

x^{2} = x (x - 1) + 3 (x - 1)

Expand

x (x - 1) : x^{2} - x

x^{2} = x^{2} - x + 3 (x - 1)

Expand

3 (x - 1) : 3 x - 3

x^{2} = x^{2} - x + 3 x - 3

Add similar elements:

- x + 3 x = 2 x

x^{2} = x^{2} + 2 x - 3

Subtract

x^{2} + 2 x

from both sides

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

Simplify

- 2 x = - 3

Divide both sides by

- 2

x = \frac{3}{2}

Verify Solutions

Find undefined (singularity) points:

x = 1, x = 0

Combine undefined points with solutions:

x = \frac{3}{2}

\frac{1}{x + 2} - 3 = \frac{10}{x + 2}

\frac{45}{5 + y} = 13

x, y = \frac{x + 3}{2 x - 8}

z + \frac{10}{z} - 6 = 0

α = \frac{λ}{α}