Solve by Factoring 1/(x-1)+1/(4x-4)=5/4

$\frac{1}{x - 1} + \frac{1}{4 x - 4} = \frac{5}{4}$

Step 1

Subtract $\frac{5}{4}$ from both sides of the equation.

$\frac{1}{x - 1} + \frac{1}{4 x - 4} - \frac{5}{4} = 0$

Step 2

Simplify $\frac{1}{x - 1} + \frac{1}{4 x - 4} - \frac{5}{4}$ .

Factor $4$ out of $4 x - 4$ .

Factor $4$ out of $4 x$ .

$\frac{1}{x - 1} + \frac{1}{4 (x) - 4} - \frac{5}{4} = 0$

Factor $4$ out of $- 4$ .

$\frac{1}{x - 1} + \frac{1}{4 (x) + 4 (- 1)} - \frac{5}{4} = 0$

Factor $4$ out of $4 (x) + 4 (- 1)$ .

$\frac{1}{x - 1} + \frac{1}{4 (x - 1)} - \frac{5}{4} = 0$

To write $\frac{1}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{x - 1} \cdot \frac{4}{4} + \frac{1}{4 (x - 1)} - \frac{5}{4} = 0$

Write each expression with a common denominator of $(x - 1) \cdot 4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x - 1}$ by $\frac{4}{4}$ .

$\frac{4}{(x - 1) \cdot 4} + \frac{1}{4 (x - 1)} - \frac{5}{4} = 0$

Reorder the factors of $(x - 1) \cdot 4$ .

$\frac{4}{4 (x - 1)} + \frac{1}{4 (x - 1)} - \frac{5}{4} = 0$

Combine the numerators over the common denominator.

$\frac{4 + 1}{4 (x - 1)} - \frac{5}{4} = 0$

Add $4$ and $1$ .

$\frac{5}{4 (x - 1)} - \frac{5}{4} = 0$

Step 3

To write $- \frac{5}{4}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{5}{4 (x - 1)} - \frac{5}{4} \cdot \frac{x - 1}{x - 1} = 0$

Step 4

Multiply $\frac{5}{4}$ by $\frac{x - 1}{x - 1}$ .

$\frac{5}{4 (x - 1)} - \frac{5 (x - 1)}{4 (x - 1)} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{5 - 5 (x - 1)}{4 (x - 1)} = 0$

Step 6

Simplify the numerator.

Factor $5$ out of $5 - 5 (x - 1)$ .

Factor $5$ out of $5$ .

$\frac{5 (1) - 5 (x - 1)}{4 (x - 1)} = 0$

Factor $5$ out of $- 5 (x - 1)$ .

$\frac{5 (1) + 5 (- (x - 1))}{4 (x - 1)} = 0$

Factor $5$ out of $5 (1) + 5 (- (x - 1))$ .

$\frac{5 (1 - (x - 1))}{4 (x - 1)} = 0$

Apply the distributive property.

$\frac{5 (1 - x + 1)}{4 (x - 1)} = 0$

Multiply $- 1$ by $- 1$ .

$\frac{5 (1 - x + 1)}{4 (x - 1)} = 0$

Add $1$ and $1$ .

$\frac{5 (- x + 2)}{4 (x - 1)} = 0$

Step 7

Set the numerator equal to zero.

$5 (- x + 2) = 0$

Step 8

Solve the equation for $x$ .

Divide each term in $5 (- x + 2) = 0$ by $5$ and simplify.

Divide each term in $5 (- x + 2) = 0$ by $5$ .

$\frac{5 (- x + 2)}{5} = \frac{0}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 (- x + 2)}{5} = \frac{0}{5}$

Divide $- x + 2$ by $1$ .

$- x + 2 = \frac{0}{5}$

Simplify the right side.

Divide $0$ by $5$ .

$- x + 2 = 0$

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

Solve by Factoring 1/(x-1)+1/(4x-4)=5/4

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