Algebra Examples

$\frac{\frac{x + 1}{x - 1} + 1}{\frac{5}{x - 1}} = 2$

Step 1

Factor each term.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$\frac{\frac{x + 1}{x - 1} + \frac{x - 1}{x - 1}}{\frac{5}{x - 1}} = 2$

Step 1.2

Combine the numerators over the common denominator.

$\frac{\frac{x + 1 + x - 1}{x - 1}}{\frac{5}{x - 1}} = 2$

Step 1.3

Reorder terms.

$\frac{\frac{x + x + 1 - 1}{x - 1}}{\frac{5}{x - 1}} = 2$

Step 1.4

Rewrite $\frac{x + x + 1 - 1}{x - 1}$ in a factored form.

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Step 1.4.1

Add $x$ and $x$ .

$\frac{\frac{2 x + 1 - 1}{x - 1}}{\frac{5}{x - 1}} = 2$

Step 1.4.2

Subtract $1$ from $1$ .

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

Step 1.4.3

Add $2 x$ and $0$ .

$\frac{\frac{2 x}{x - 1}}{\frac{5}{x - 1}} = 2$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x}{x - 1} \cdot \frac{x - 1}{5} = 2$

Step 1.6

Cancel the common factor of $x - 1$ .

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Step 1.6.1

Cancel the common factor.

$\frac{2 x}{x - 1} \cdot \frac{x - 1}{5} = 2$

Step 1.6.2

Rewrite the expression.

$2 x \frac{1}{5} = 2$

Step 1.7

Combine $2$ and $\frac{1}{5}$ .

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

Step 1.8

Combine $x$ and $\frac{2}{5}$ .

$\frac{x \cdot 2}{5} = 2$

Step 1.9

Move $2$ to the left of $x$ .

$\frac{2 \cdot x}{5} = 2$

Step 1.10

Multiply $2$ by $x$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$5, 1$

Step 2.2

The LCM of one and any expression is the expression.

$5$

Step 3

Multiply each term in $\frac{2 x}{5} = 2$ by $5$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{2 x}{5} = 2$ by $5$ .

$\frac{2 x}{5} \cdot 5 = 2 \cdot 5$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{5} \cdot 5 = 2 \cdot 5$

Step 3.2.1.2

Rewrite the expression.

$2 x = 2 \cdot 5$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $2$ by $5$ .

$2 x = 10$

Step 4

Divide each term in $2 x = 10$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = 10$ by $2$ .

$\frac{2 x}{2} = \frac{10}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{10}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $10$ by $2$ .

$x = 5$