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

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

Move $\frac{5}{4}$ to the left side of the equation by subtracting it from both sides.

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

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

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

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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 $1$ .

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

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

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 $1$ .

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

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

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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 $1$ .

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

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

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 $1$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

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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 $1$ .

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

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$4 (x - 1), 1$

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $4, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2$

Multiply $2$ by $2$ .

$4$

The factor for $x - 1$ is $x - 1$ itself.

$(x - 1) = x - 1$

$(x - 1)$ occurs $1$ time.

The LCM of $x - 1$ is the result of multiplying all factors the greatest number of times they occur in either term.

$x - 1$

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$4 (x - 1)$

Multiply each term by $4 (x - 1)$ and simplify.

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Multiply each term in $\frac{5 (- x + 2)}{4 (x - 1)} = 0$ by $4 (x - 1)$ in order to remove all the denominators from the equation.

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

Simplify $\frac{5 (- x + 2)}{4 (x - 1)} \cdot (4 (x - 1))$ .

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Rewrite using the commutative property of multiplication.

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

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Cancel the common factor.

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

Rewrite the expression.

$5 (- x + 2) = 0 \cdot (4 (x - 1))$

Apply the distributive property.

$5 (- x) + 5 \cdot 2 = 0 \cdot (4 (x - 1))$

Multiply.

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Multiply $- 1$ by $5$ .

$- 5 x + 5 \cdot 2 = 0 \cdot (4 (x - 1))$

Multiply $5$ by $2$ .

$- 5 x + 10 = 0 \cdot (4 (x - 1))$

Simplify $0 \cdot (4 (x - 1))$ .

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Apply the distributive property.

$- 5 x + 10 = 0 \cdot (4 x + 4 \cdot - 1)$

Multiply.

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Multiply $4$ by $- 1$ .

$- 5 x + 10 = 0 \cdot (4 x - 4)$

Multiply $0$ by $4 x - 4$ .

$- 5 x + 10 = 0$

Solve the equation.

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Subtract $10$ from both sides of the equation.

$- 5 x = - 10$

Divide each term by $- 5$ and simplify.

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Divide each term in $- 5 x = - 10$ by $- 5$ .

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

Cancel the common factor of $- 5$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

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

Divide $- 10$ by $- 5$ .

$x = 2$

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

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