Algebra Examples

$\frac{6}{x + 5} = 1 - \frac{1}{x - 5}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $1$ from both sides of the equation.

$\frac{6}{x + 5} - 1 = - \frac{1}{x - 5}$

Step 1.2

Add $\frac{1}{x - 5}$ to both sides of the equation.

$\frac{6}{x + 5} - 1 + \frac{1}{x - 5} = 0$

Step 2

Simplify $\frac{6}{x + 5} - 1 + \frac{1}{x - 5}$ .

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Step 2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{6}{x + 5} - 1 \cdot \frac{x + 5}{x + 5} + \frac{1}{x - 5} = 0$

Step 2.2

Combine $- 1$ and $\frac{x + 5}{x + 5}$ .

$\frac{6}{x + 5} + \frac{- (x + 5)}{x + 5} + \frac{1}{x - 5} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{6 - (x + 5)}{x + 5} + \frac{1}{x - 5} = 0$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Apply the distributive property.

$\frac{6 - x - 1 \cdot 5}{x + 5} + \frac{1}{x - 5} = 0$

Step 2.4.2

Multiply $- 1$ by $5$ .

$\frac{6 - x - 5}{x + 5} + \frac{1}{x - 5} = 0$

Step 2.4.3

Subtract $5$ from $6$ .

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

Step 2.5

To write $\frac{- x + 1}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

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

Step 2.6

To write $\frac{1}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

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

Step 2.7

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

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Step 2.7.1

Multiply $\frac{- x + 1}{x + 5}$ by $\frac{x - 5}{x - 5}$ .

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

Step 2.7.2

Multiply $\frac{1}{x - 5}$ by $\frac{x + 5}{x + 5}$ .

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

Step 2.7.3

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Expand $(- x + 1) (x - 5)$ using the FOIL Method.

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Step 2.9.1.1

Apply the distributive property.

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

Step 2.9.1.2

Apply the distributive property.

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

Step 2.9.1.3

Apply the distributive property.

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

Step 2.9.2

Simplify and combine like terms.

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Step 2.9.2.1

Simplify each term.

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Step 2.9.2.1.1

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

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Step 2.9.2.1.1.1

Move $x$ .

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

Step 2.9.2.1.1.2

Multiply $x$ by $x$ .

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

Step 2.9.2.1.2

Multiply $- 5$ by $- 1$ .

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

Step 2.9.2.1.3

Multiply $x$ by $1$ .

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

Step 2.9.2.1.4

Multiply $- 5$ by $1$ .

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

Step 2.9.2.2

Add $5 x$ and $x$ .

$\frac{- x^{2} + 6 x - 5 + x + 5}{(x + 5) (x - 5)} = 0$

Step 2.9.3

Add $6 x$ and $x$ .

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

Step 2.9.4

Add $- 5$ and $5$ .

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

Step 2.9.5

Add $- x^{2} + 7 x$ and $0$ .

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

Step 2.9.6

Factor $x$ out of $- x^{2} + 7 x$ .

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Step 2.9.6.1

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

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

Step 2.9.6.2

Factor $x$ out of $7 x$ .

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

Step 2.9.6.3

Factor $x$ out of $x (- x) + x \cdot 7$ .

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

Step 2.10

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

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

Step 2.11

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{x (- (x) - 1 (- 7))}{(x + 5) (x - 5)} = 0$

Step 2.12

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

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

Step 2.13

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$\frac{x (- 1 (x - 7))}{(x + 5) (x - 5)} = 0$

Step 2.14

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$x (x - 7) = 0$

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 $0$ .

$x = 0$

$x - 7 = 0$

Step 4.2

Set $x$ equal to $0$ .

$x = 0$

Step 4.3

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

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Step 4.3.1

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

$x - 7 = 0$

Step 4.3.2

Add $7$ to both sides of the equation.

$x = 7$

Step 4.4

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

$x = 0, 7$