Algebra Examples

$\frac{5}{x + 2} + \frac{2}{x - 5} = \frac{14}{(x + 2) (x - 5)}$

Step 1

Subtract $\frac{14}{(x + 2) (x - 5)}$ from both sides of the equation.

$\frac{5}{x + 2} + \frac{2}{x - 5} - \frac{14}{(x + 2) (x - 5)} = 0$

Step 2

Simplify $\frac{5}{x + 2} + \frac{2}{x - 5} - \frac{14}{(x + 2) (x - 5)}$ .

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Step 2.1

To write $\frac{5}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{5}{x + 2} \cdot \frac{x - 5}{x - 5} + \frac{2}{x - 5} - \frac{14}{(x + 2) (x - 5)} = 0$

Step 2.2

To write $\frac{2}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{5}{x + 2} \cdot \frac{x - 5}{x - 5} + \frac{2}{x - 5} \cdot \frac{x + 2}{x + 2} - \frac{14}{(x + 2) (x - 5)} = 0$

Step 2.3

Write each expression with a common denominator of $(x + 2) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{5}{x + 2}$ by $\frac{x - 5}{x - 5}$ .

$\frac{5 (x - 5)}{(x + 2) (x - 5)} + \frac{2}{x - 5} \cdot \frac{x + 2}{x + 2} - \frac{14}{(x + 2) (x - 5)} = 0$

Step 2.3.2

Multiply $\frac{2}{x - 5}$ by $\frac{x + 2}{x + 2}$ .

$\frac{5 (x - 5)}{(x + 2) (x - 5)} + \frac{2 (x + 2)}{(x - 5) (x + 2)} - \frac{14}{(x + 2) (x - 5)} = 0$

Step 2.3.3

Reorder the factors of $(x - 5) (x + 2)$ .

$\frac{5 (x - 5)}{(x + 2) (x - 5)} + \frac{2 (x + 2)}{(x + 2) (x - 5)} - \frac{14}{(x + 2) (x - 5)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{5 (x - 5) + 2 (x + 2)}{(x + 2) (x - 5)} - \frac{14}{(x + 2) (x - 5)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{5 (x - 5) + 2 (x + 2) - 14}{(x + 2) (x - 5)} = 0$

Step 2.6

Simplify each term.

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Step 2.6.1

Apply the distributive property.

$\frac{5 x + 5 \cdot - 5 + 2 (x + 2) - 14}{(x + 2) (x - 5)} = 0$

Step 2.6.2

Multiply $5$ by $- 5$ .

$\frac{5 x - 25 + 2 (x + 2) - 14}{(x + 2) (x - 5)} = 0$

Step 2.6.3

Apply the distributive property.

$\frac{5 x - 25 + 2 x + 2 \cdot 2 - 14}{(x + 2) (x - 5)} = 0$

Step 2.6.4

Multiply $2$ by $2$ .

$\frac{5 x - 25 + 2 x + 4 - 14}{(x + 2) (x - 5)} = 0$

Step 2.7

Add $5 x$ and $2 x$ .

$\frac{7 x - 25 + 4 - 14}{(x + 2) (x - 5)} = 0$

Step 2.8

Add $- 25$ and $4$ .

$\frac{7 x - 21 - 14}{(x + 2) (x - 5)} = 0$

Step 2.9

Subtract $14$ from $- 21$ .

$\frac{7 x - 35}{(x + 2) (x - 5)} = 0$

Step 2.10

Factor $7$ out of $7 x - 35$ .

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Step 2.10.1

Factor $7$ out of $7 x$ .

$\frac{7 (x) - 35}{(x + 2) (x - 5)} = 0$

Step 2.10.2

Factor $7$ out of $- 35$ .

$\frac{7 x + 7 \cdot - 5}{(x + 2) (x - 5)} = 0$

Step 2.10.3

Factor $7$ out of $7 x + 7 \cdot - 5$ .

$\frac{7 (x - 5)}{(x + 2) (x - 5)} = 0$

Step 2.11

Cancel the common factor of $x - 5$ .

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Step 2.11.1

Cancel the common factor.

$\frac{7 (x - 5)}{(x + 2) (x - 5)} = 0$

Step 2.11.2

Rewrite the expression.

$\frac{7}{x + 2} = 0$

Step 3

Set the denominator in $\frac{7}{x + 2}$ equal to $0$ to find where the expression is undefined.

$x + 2 = 0$

Step 4

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5