Algebra Examples

$\frac{1}{x - 4} + 1 = \frac{14}{x + 2}$

Step 1

Subtract $\frac{14}{x + 2}$ from both sides of the equation.

$\frac{1}{x - 4} + 1 - \frac{14}{x + 2} = 0$

Step 2

Simplify $\frac{1}{x - 4} + 1 - \frac{14}{x + 2}$ .

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Step 2.1

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

$\frac{1}{x - 4} + \frac{x - 4}{x - 4} - \frac{14}{x + 2} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{1 + x - 4}{x - 4} - \frac{14}{x + 2} = 0$

Step 2.3

Subtract $4$ from $1$ .

$\frac{x - 3}{x - 4} - \frac{14}{x + 2} = 0$

Step 2.4

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

$\frac{x - 3}{x - 4} \cdot \frac{x + 2}{x + 2} - \frac{14}{x + 2} = 0$

Step 2.5

To write $- \frac{14}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{x - 3}{x - 4} \cdot \frac{x + 2}{x + 2} - \frac{14}{x + 2} \cdot \frac{x - 4}{x - 4} = 0$

Step 2.6

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 2.6.1

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

$\frac{(x - 3) (x + 2)}{(x - 4) (x + 2)} - \frac{14}{x + 2} \cdot \frac{x - 4}{x - 4} = 0$

Step 2.6.2

Multiply $\frac{14}{x + 2}$ by $\frac{x - 4}{x - 4}$ .

$\frac{(x - 3) (x + 2)}{(x - 4) (x + 2)} - \frac{14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.6.3

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

$\frac{(x - 3) (x + 2)}{(x + 2) (x - 4)} - \frac{14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.7

Combine the numerators over the common denominator.

$\frac{(x - 3) (x + 2) - 14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Expand $(x - 3) (x + 2)$ using the FOIL Method.

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Step 2.8.1.1

Apply the distributive property.

$\frac{x (x + 2) - 3 (x + 2) - 14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.8.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - 3 (x + 2) - 14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.8.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 2 - 3 x - 3 \cdot 2 - 14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.8.2

Simplify and combine like terms.

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Step 2.8.2.1

Simplify each term.

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Step 2.8.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 2 - 3 x - 3 \cdot 2 - 14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.8.2.1.2

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

$\frac{x^{2} + 2 \cdot x - 3 x - 3 \cdot 2 - 14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.8.2.1.3

Multiply $- 3$ by $2$ .

$\frac{x^{2} + 2 x - 3 x - 6 - 14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.8.2.2

Subtract $3 x$ from $2 x$ .

$\frac{x^{2} - x - 6 - 14 (x - 4)}{(x + 2) (x - 4)} = 0$

Step 2.8.3

Apply the distributive property.

$\frac{x^{2} - x - 6 - 14 x - 14 \cdot - 4}{(x + 2) (x - 4)} = 0$

Step 2.8.4

Multiply $- 14$ by $- 4$ .

$\frac{x^{2} - x - 6 - 14 x + 56}{(x + 2) (x - 4)} = 0$

Step 2.8.5

Subtract $14 x$ from $- x$ .

$\frac{x^{2} - 15 x - 6 + 56}{(x + 2) (x - 4)} = 0$

Step 2.8.6

Add $- 6$ and $56$ .

$\frac{x^{2} - 15 x + 50}{(x + 2) (x - 4)} = 0$

Step 2.8.7

Factor $x^{2} - 15 x + 50$ using the AC method.

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Step 2.8.7.1

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 $50$ and whose sum is $- 15$ .

$- 10, - 5$

Step 2.8.7.2

Write the factored form using these integers.

$\frac{(x - 10) (x - 5)}{(x + 2) (x - 4)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 10) (x - 5) = 0$

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 $0$ .

$x - 10 = 0$

$x - 5 = 0$

Step 4.2

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 4.2.2

Add $10$ to both sides of the equation.

$x = 10$

Step 4.3

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.3.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.4

The final solution is all the values that make $(x - 10) (x - 5) = 0$ true.

$x = 10, 5$