Algebra Examples

$\frac{6}{x - 4} = \frac{x - 3}{x + 2}$

Step 1

Subtract $\frac{x - 3}{x + 2}$ from both sides of the equation.

$\frac{6}{x - 4} - \frac{x - 3}{x + 2} = 0$

Step 2

Simplify $\frac{6}{x - 4} - \frac{x - 3}{x + 2}$ .

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Step 2.1

To write $\frac{6}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{6}{x - 4} \cdot \frac{x + 2}{x + 2} - \frac{x - 3}{x + 2} = 0$

Step 2.2

To write $- \frac{x - 3}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{6}{x - 4} \cdot \frac{x + 2}{x + 2} - \frac{x - 3}{x + 2} \cdot \frac{x - 4}{x - 4} = 0$

Step 2.3

Write each expression with a common denominator of $(x - 4) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{6}{x - 4}$ by $\frac{x + 2}{x + 2}$ .

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

Step 2.3.2

Multiply $\frac{x - 3}{x + 2}$ by $\frac{x - 4}{x - 4}$ .

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

Step 2.3.3

Reorder the factors of $(x - 4) (x + 2)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Multiply $6$ by $2$ .

$\frac{6 x + 12 - (x - 3) (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.5.3

Apply the distributive property.

$\frac{6 x + 12 + (- x - - 3) (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.5.4

Multiply $- 1$ by $- 3$ .

$\frac{6 x + 12 + (- x + 3) (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.5.5

Expand $(- x + 3) (x - 4)$ using the FOIL Method.

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Step 2.5.5.1

Apply the distributive property.

$\frac{6 x + 12 - x (x - 4) + 3 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.5.5.2

Apply the distributive property.

$\frac{6 x + 12 - x \cdot x - x \cdot - 4 + 3 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.5.5.3

Apply the distributive property.

$\frac{6 x + 12 - x \cdot x - x \cdot - 4 + 3 x + 3 \cdot - 4}{(x + 2) (x - 4)} = 0$

Step 2.5.6

Simplify and combine like terms.

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Step 2.5.6.1

Simplify each term.

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Step 2.5.6.1.1

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

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Step 2.5.6.1.1.1

Move $x$ .

$\frac{6 x + 12 - (x \cdot x) - x \cdot - 4 + 3 x + 3 \cdot - 4}{(x + 2) (x - 4)} = 0$

Step 2.5.6.1.1.2

Multiply $x$ by $x$ .

$\frac{6 x + 12 - x^{2} - x \cdot - 4 + 3 x + 3 \cdot - 4}{(x + 2) (x - 4)} = 0$

Step 2.5.6.1.2

Multiply $- 4$ by $- 1$ .

$\frac{6 x + 12 - x^{2} + 4 x + 3 x + 3 \cdot - 4}{(x + 2) (x - 4)} = 0$

Step 2.5.6.1.3

Multiply $3$ by $- 4$ .

$\frac{6 x + 12 - x^{2} + 4 x + 3 x - 12}{(x + 2) (x - 4)} = 0$

Step 2.5.6.2

Add $4 x$ and $3 x$ .

$\frac{6 x + 12 - x^{2} + 7 x - 12}{(x + 2) (x - 4)} = 0$

Step 2.5.7

Add $6 x$ and $7 x$ .

$\frac{13 x + 12 - x^{2} - 12}{(x + 2) (x - 4)} = 0$

Step 2.5.8

Subtract $12$ from $12$ .

$\frac{13 x - x^{2} + 0}{(x + 2) (x - 4)} = 0$

Step 2.5.9

Add $13 x - x^{2}$ and $0$ .

$\frac{13 x - x^{2}}{(x + 2) (x - 4)} = 0$

Step 2.5.10

Factor $x$ out of $13 x - x^{2}$ .

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Step 2.5.10.1

Factor $x$ out of $13 x$ .

$\frac{x \cdot 13 - x^{2}}{(x + 2) (x - 4)} = 0$

Step 2.5.10.2

Factor $x$ out of $- x^{2}$ .

$\frac{x \cdot 13 + x (- x)}{(x + 2) (x - 4)} = 0$

Step 2.5.10.3

Factor $x$ out of $x \cdot 13 + x (- x)$ .

$\frac{x (13 - x)}{(x + 2) (x - 4)} = 0$

Step 3

Set the numerator equal to zero.

$x (13 - x) = 0$

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 $0$ .

$x = 0$

$13 - x = 0$

Step 4.2

Set $x$ equal to $0$ .

$x = 0$

Step 4.3

Set $13 - x$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $13 - x$ equal to $0$ .

$13 - x = 0$

Step 4.3.2

Solve $13 - x = 0$ for $x$ .

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Step 4.3.2.1

Subtract $13$ from both sides of the equation.

$- x = - 13$

Step 4.3.2.2

Divide each term in $- x = - 13$ by $- 1$ and simplify.

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Step 4.3.2.2.1

Divide each term in $- x = - 13$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 13}{- 1}$

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.

$\frac{x}{1} = \frac{- 13}{- 1}$

Step 4.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 13}{- 1}$

Step 4.3.2.2.3

Simplify the right side.

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Step 4.3.2.2.3.1

Divide $- 13$ by $- 1$ .

$x = 13$

Step 4.4

The final solution is all the values that make $x (13 - x) = 0$ true.

$x = 0, 13$