What is the answer to ((x-1))/((2x-1))+((2x-1))/((x-1))= 5/2 ?

The answer to ((x-1))/((2x-1))+((2x-1))/((x-1))= 5/2 is x= 1/3

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((x-1))/((2x-1))+((2x-1))/((x-1))= 5/2

\frac{( x - 1 )}{( 2 x - 1 )} + \frac{( 2 x - 1 )}{( x - 1 )} = \frac{5}{2}

x = \frac{1}{3}

Decimal Notation

x = 0.33333 \dots

Solution steps

\frac{( x - 1 )}{( 2 x - 1 )} + \frac{( 2 x - 1 )}{( x - 1 )} = \frac{5}{2}

Multiply by LCM

2 (x - 1)^{2} + 2 (2 x - 1)^{2} = 5 (x - 1) (2 x - 1)

Expand

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

10 x^{2} - 12 x + 4 = 5 (2 x^{2} - 3 x + 1)

Expand

5 (2 x^{2} - 3 x + 1) : 10 x^{2} - 15 x + 5

10 x^{2} - 12 x + 4 = 10 x^{2} - 15 x + 5

Expand

2 (x - 1)^{2} + 2 (2 x - 1)^{2} : 10 x^{2} - 12 x + 4

10 x^{2} - 12 x + 4 = 10 x^{2} - 15 x + 5

Move

4

to the right side

10 x^{2} - 12 x = 10 x^{2} - 15 x + 1

Subtract

10 x^{2} - 15 x

from both sides

10 x^{2} - 12 x - (10 x^{2} - 15 x) = 10 x^{2} - 15 x + 1 - (10 x^{2} - 15 x)

Simplify

3 x = 1

Divide both sides by

3

x = \frac{1}{3}

Verify Solutions

Find undefined (singularity) points:

x = \frac{1}{2}, x = 1

Combine undefined points with solutions:

x = \frac{1}{3}

\frac{1}{4 x - 4} = 4

(\frac{x ^{2} + 1}{x ^{2}}) - 4 (\frac{x - 1}{x}) - 14 = 0

f (n) = \frac{w ^{n + 1}}{w ^{2 n}}

\frac{x ^{2} - 1}{2 x - 4} = 0

\frac{( ( 2 n - 1 ) ! )}{( ( 2 n - 3 ) ! )} = 120

What is the answer to ((x-1))/((2x-1))+((2x-1))/((x-1))= 5/2 ?
The answer to ((x-1))/((2x-1))+((2x-1))/((x-1))= 5/2 is x= 1/3
Solve for x: ((x-1))/((2x-1))+((2x-1))/((x-1))= 5/2
The solution is x= 1/3