Algebra Examples

$\frac{3}{x + 2} - \frac{5}{x} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{3}{x + 2} - \frac{5}{x} - 1 = 0$

Step 2

Simplify $\frac{3}{x + 2} - \frac{5}{x} - 1$ .

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Step 2.1

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

$\frac{3}{x + 2} \cdot \frac{x}{x} - \frac{5}{x} - 1 = 0$

Step 2.2

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

$\frac{3}{x + 2} \cdot \frac{x}{x} - \frac{5}{x} \cdot \frac{x + 2}{x + 2} - 1 = 0$

Step 2.3

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

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Step 2.3.1

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

$\frac{3 x}{(x + 2) x} - \frac{5}{x} \cdot \frac{x + 2}{x + 2} - 1 = 0$

Step 2.3.2

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

$\frac{3 x}{(x + 2) x} - \frac{5 (x + 2)}{x (x + 2)} - 1 = 0$

Step 2.3.3

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

$\frac{3 x}{x (x + 2)} - \frac{5 (x + 2)}{x (x + 2)} - 1 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x - 5 (x + 2)}{x (x + 2)} - 1 = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{3 x - 5 x - 5 \cdot 2}{x (x + 2)} - 1 = 0$

Step 2.5.2

Multiply $- 5$ by $2$ .

$\frac{3 x - 5 x - 10}{x (x + 2)} - 1 = 0$

Step 2.5.3

Subtract $5 x$ from $3 x$ .

$\frac{- 2 x - 10}{x (x + 2)} - 1 = 0$

Step 2.5.4

Factor $2$ out of $- 2 x - 10$ .

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Step 2.5.4.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x) - 10}{x (x + 2)} - 1 = 0$

Step 2.5.4.2

Factor $2$ out of $- 10$ .

$\frac{2 (- x) + 2 (- 5)}{x (x + 2)} - 1 = 0$

Step 2.5.4.3

Factor $2$ out of $2 (- x) + 2 (- 5)$ .

$\frac{2 (- x - 5)}{x (x + 2)} - 1 = 0$

Step 2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x (x + 2)}{x (x + 2)}$ .

$\frac{2 (- x - 5)}{x (x + 2)} - 1 \cdot \frac{x (x + 2)}{x (x + 2)} = 0$

Step 2.7

Combine $- 1$ and $\frac{x (x + 2)}{x (x + 2)}$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

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

Step 2.9.2

Multiply $- 1$ by $2$ .

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

Step 2.9.3

Multiply $2$ by $- 5$ .

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

Step 2.9.4

Apply the distributive property.

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

Step 2.9.5

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

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Step 2.9.5.1

Move $x$ .

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

Step 2.9.5.2

Multiply $x$ by $x$ .

$\frac{- 2 x - 10 - x^{2} - x \cdot 2}{x (x + 2)} = 0$

Step 2.9.6

Multiply $2$ by $- 1$ .

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

Step 2.9.7

Subtract $2 x$ from $- 2 x$ .

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

Step 2.9.8

Reorder terms.

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

Step 2.10

Factor $- 1$ out of $- x^{2}$ .

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

Step 2.11

Factor $- 1$ out of $- 4 x$ .

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

Step 2.12

Factor $- 1$ out of $- (x^{2}) - (4 x)$ .

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

Step 2.13

Rewrite $- 10$ as $- 1 (10)$ .

$\frac{- (x^{2} + 4 x) - 1 (10)}{x (x + 2)} = 0$

Step 2.14

Factor $- 1$ out of $- (x^{2} + 4 x) - 1 (10)$ .

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

Step 2.15

Rewrite $- (x^{2} + 4 x + 10)$ as $- 1 (x^{2} + 4 x + 10)$ .

$\frac{- 1 (x^{2} + 4 x + 10)}{x (x + 2)} = 0$

Step 2.16

Move the negative in front of the fraction.

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

Step 3

Set the denominator in $\frac{x^{2} + 4 x + 10}{x (x + 2)}$ equal to $0$ to find where the expression is undefined.

$x (x + 2) = 0$

Step 4

Solve 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 $0$ .

$x = 0$

$x + 2 = 0$

Step 4.2

Set $x$ equal to $0$ .

$x = 0$

Step 4.3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.4

The final solution is all the values that make $x (x + 2) = 0$ true.

$x = 0, - 2$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 2, x = 0$

Step 6