Algebra Examples

$\frac{4}{x - 1} = \frac{x + 1}{12}$

Step 1

Subtract $\frac{x + 1}{12}$ from both sides of the equation.

$\frac{4}{x - 1} - \frac{x + 1}{12} = 0$

Step 2

Simplify $\frac{4}{x - 1} - \frac{x + 1}{12}$ .

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Step 2.1

To write $\frac{4}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$\frac{4}{x - 1} \cdot \frac{12}{12} - \frac{x + 1}{12} = 0$

Step 2.2

To write $- \frac{x + 1}{12}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{4}{x - 1} \cdot \frac{12}{12} - \frac{x + 1}{12} \cdot \frac{x - 1}{x - 1} = 0$

Step 2.3

Write each expression with a common denominator of $(x - 1) \cdot 12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{4}{x - 1}$ by $\frac{12}{12}$ .

$\frac{4 \cdot 12}{(x - 1) \cdot 12} - \frac{x + 1}{12} \cdot \frac{x - 1}{x - 1} = 0$

Step 2.3.2

Multiply $\frac{x + 1}{12}$ by $\frac{x - 1}{x - 1}$ .

$\frac{4 \cdot 12}{(x - 1) \cdot 12} - \frac{(x + 1) (x - 1)}{12 (x - 1)} = 0$

Step 2.3.3

Reorder the factors of $(x - 1) \cdot 12$ .

$\frac{4 \cdot 12}{12 (x - 1)} - \frac{(x + 1) (x - 1)}{12 (x - 1)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{4 \cdot 12 - (x + 1) (x - 1)}{12 (x - 1)} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $4$ by $12$ .

$\frac{48 - (x + 1) (x - 1)}{12 (x - 1)} = 0$

Step 2.5.2

Apply the distributive property.

$\frac{48 + (- x - 1 \cdot 1) (x - 1)}{12 (x - 1)} = 0$

Step 2.5.3

Multiply $- 1$ by $1$ .

$\frac{48 + (- x - 1) (x - 1)}{12 (x - 1)} = 0$

Step 2.5.4

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

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Step 2.5.4.1

Apply the distributive property.

$\frac{48 - x (x - 1) - 1 (x - 1)}{12 (x - 1)} = 0$

Step 2.5.4.2

Apply the distributive property.

$\frac{48 - x \cdot x - x \cdot - 1 - 1 (x - 1)}{12 (x - 1)} = 0$

Step 2.5.4.3

Apply the distributive property.

$\frac{48 - x \cdot x - x \cdot - 1 - 1 x - 1 \cdot - 1}{12 (x - 1)} = 0$

Step 2.5.5

Simplify and combine like terms.

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Step 2.5.5.1

Simplify each term.

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Step 2.5.5.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.5.1.1.1

Move $x$ .

$\frac{48 - (x \cdot x) - x \cdot - 1 - 1 x - 1 \cdot - 1}{12 (x - 1)} = 0$

Step 2.5.5.1.1.2

Multiply $x$ by $x$ .

$\frac{48 - x^{2} - x \cdot - 1 - 1 x - 1 \cdot - 1}{12 (x - 1)} = 0$

Step 2.5.5.1.2

Multiply $- x \cdot - 1$ .

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Step 2.5.5.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{48 - x^{2} + 1 x - 1 x - 1 \cdot - 1}{12 (x - 1)} = 0$

Step 2.5.5.1.2.2

Multiply $x$ by $1$ .

$\frac{48 - x^{2} + x - 1 x - 1 \cdot - 1}{12 (x - 1)} = 0$

Step 2.5.5.1.3

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

$\frac{48 - x^{2} + x - x - 1 \cdot - 1}{12 (x - 1)} = 0$

Step 2.5.5.1.4

Multiply $- 1$ by $- 1$ .

$\frac{48 - x^{2} + x - x + 1}{12 (x - 1)} = 0$

Step 2.5.5.2

Subtract $x$ from $x$ .

$\frac{48 - x^{2} + 0 + 1}{12 (x - 1)} = 0$

Step 2.5.5.3

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

$\frac{48 - x^{2} + 1}{12 (x - 1)} = 0$

Step 2.5.6

Add $48$ and $1$ .

$\frac{- x^{2} + 49}{12 (x - 1)} = 0$

Step 2.5.7

Rewrite $- x^{2} + 49$ in a factored form.

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Step 2.5.7.1

Rewrite $49$ as $7^{2}$ .

$\frac{- x^{2} + 7^{2}}{12 (x - 1)} = 0$

Step 2.5.7.2

Reorder $- x^{2}$ and $7^{2}$ .

$\frac{7^{2} - x^{2}}{12 (x - 1)} = 0$

Step 2.5.7.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7$ and $b = x$ .

$\frac{(7 + x) (7 - x)}{12 (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$(7 + x) (7 - x) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$7 + x = 0$

$7 - x = 0$

Step 4.2

Set $7 + x$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $7 + x$ equal to $0$ .

$7 + x = 0$

Step 4.2.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 4.3

Set $7 - x$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $7 - x$ equal to $0$ .

$7 - x = 0$

Step 4.3.2

Solve $7 - x = 0$ for $x$ .

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Step 4.3.2.1

Subtract $7$ from both sides of the equation.

$- x = - 7$

Step 4.3.2.2

Divide each term in $- x = - 7$ by $- 1$ and simplify.

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Step 4.3.2.2.1

Divide each term in $- x = - 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 7}{- 1}$

Step 4.3.2.2.2

Simplify the left side.

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Step 4.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 7}{- 1}$

Step 4.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 7}{- 1}$

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Divide $- 7$ by $- 1$ .

$x = 7$

Step 4.4

The final solution is all the values that make $(7 + x) (7 - x) = 0$ true.

$x = - 7, 7$