Algebra Examples

$1 - \frac{3}{x - 1} = - \frac{6}{x^{2} - 1}$

Step 1

Add $\frac{6}{x^{2} - 1}$ to both sides of the equation.

$1 - \frac{3}{x - 1} + \frac{6}{x^{2} - 1} = 0$

Step 2

Simplify $1 - \frac{3}{x - 1} + \frac{6}{x^{2} - 1}$ .

Tap for more steps...

Step 2.1

Simplify the denominator.

Tap for more steps...

Step 2.1.1

Rewrite $1$ as $1^{2}$ .

$1 - \frac{3}{x - 1} + \frac{6}{x^{2} - 1^{2}} = 0$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$1 - \frac{3}{x - 1} + \frac{6}{(x + 1) (x - 1)} = 0$

Step 2.2

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

$\frac{x - 1}{x - 1} - \frac{3}{x - 1} + \frac{6}{(x + 1) (x - 1)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{x - 1 - 3}{x - 1} + \frac{6}{(x + 1) (x - 1)} = 0$

Step 2.4

Subtract $3$ from $- 1$ .

$\frac{x - 4}{x - 1} + \frac{6}{(x + 1) (x - 1)} = 0$

Step 2.5

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

$\frac{x - 4}{x - 1} \cdot \frac{x + 1}{x + 1} + \frac{6}{(x + 1) (x - 1)} = 0$

Step 2.6

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

Tap for more steps...

Step 2.6.1

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

$\frac{(x - 4) (x + 1)}{(x - 1) (x + 1)} + \frac{6}{(x + 1) (x - 1)} = 0$

Step 2.6.2

Reorder the factors of $(x - 1) (x + 1)$ .

$\frac{(x - 4) (x + 1)}{(x + 1) (x - 1)} + \frac{6}{(x + 1) (x - 1)} = 0$

Step 2.7

Combine the numerators over the common denominator.

$\frac{(x - 4) (x + 1) + 6}{(x + 1) (x - 1)} = 0$

Step 2.8

Simplify the numerator.

Tap for more steps...

Step 2.8.1

Expand $(x - 4) (x + 1)$ using the FOIL Method.

Tap for more steps...

Step 2.8.1.1

Apply the distributive property.

$\frac{x (x + 1) - 4 (x + 1) + 6}{(x + 1) (x - 1)} = 0$

Step 2.8.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 - 4 (x + 1) + 6}{(x + 1) (x - 1)} = 0$

Step 2.8.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 - 4 x - 4 \cdot 1 + 6}{(x + 1) (x - 1)} = 0$

Step 2.8.2

Simplify and combine like terms.

Tap for more steps...

Step 2.8.2.1

Simplify each term.

Tap for more steps...

Step 2.8.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 1 - 4 x - 4 \cdot 1 + 6}{(x + 1) (x - 1)} = 0$

Step 2.8.2.1.2

Multiply $x$ by $1$ .

$\frac{x^{2} + x - 4 x - 4 \cdot 1 + 6}{(x + 1) (x - 1)} = 0$

Step 2.8.2.1.3

Multiply $- 4$ by $1$ .

$\frac{x^{2} + x - 4 x - 4 + 6}{(x + 1) (x - 1)} = 0$

Step 2.8.2.2

Subtract $4 x$ from $x$ .

$\frac{x^{2} - 3 x - 4 + 6}{(x + 1) (x - 1)} = 0$

Step 2.8.3

Add $- 4$ and $6$ .

$\frac{x^{2} - 3 x + 2}{(x + 1) (x - 1)} = 0$

Step 2.8.4

Factor $x^{2} - 3 x + 2$ using the AC method.

Tap for more steps...

Step 2.8.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 2.8.4.2

Write the factored form using these integers.

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

Step 2.9

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 2.9.1

Cancel the common factor.

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

Step 2.9.2

Rewrite the expression.

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

Step 3

Set the denominator in $\frac{x - 2}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5