Algebra Examples

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

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$x - 1, 1, 1$

Step 1.2

Remove parentheses.

$x - 1, 1, 1$

Step 1.3

The LCM of one and any expression is the expression.

$x - 1$

Step 2

Multiply each term in $\frac{3}{x - 1} = x + 1$ by $x - 1$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{3}{x - 1} = x + 1$ by $x - 1$ .

$\frac{3}{x - 1} (x - 1) = x (x - 1) + 1 (x - 1)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x - 1$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3}{x - 1} (x - 1) = x (x - 1) + 1 (x - 1)$

Step 2.2.1.2

Rewrite the expression.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Apply the distributive property.

$3 = x \cdot x + x \cdot - 1 + 1 (x - 1)$

Step 2.3.1.2

Multiply $x$ by $x$ .

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

Step 2.3.1.3

Move $- 1$ to the left of $x$ .

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

Step 2.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.3.1.5

Multiply $x - 1$ by $1$ .

$3 = x^{2} - x + x - 1$

Step 2.3.2

Combine the opposite terms in $x^{2} - x + x - 1$ .

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Step 2.3.2.1

Add $- x$ and $x$ .

$3 = x^{2} + 0 - 1$

Step 2.3.2.2

Add $x^{2}$ and $0$ .

$3 = x^{2} - 1$

Step 3

Solve the equation.

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Step 3.1

Rewrite the equation as $x^{2} - 1 = 3$ .

$x^{2} - 1 = 3$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1

Add $1$ to both sides of the equation.

$x^{2} = 3 + 1$

Step 3.2.2

Add $3$ and $1$ .

$x^{2} = 4$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 3.4

Simplify $\pm \sqrt{4}$ .

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Step 3.4.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$