Precalculus Examples

$\frac{x - 9}{x + 5} = \frac{x + 4}{x + 8}$

Step 1

Subtract $\frac{x + 4}{x + 8}$ from both sides of the equation.

$\frac{x - 9}{x + 5} - \frac{x + 4}{x + 8} = 0$

Step 2

Simplify $\frac{x - 9}{x + 5} - \frac{x + 4}{x + 8}$ .

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Step 2.1

To write $\frac{x - 9}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x + 8}{x + 8}$ .

$\frac{x - 9}{x + 5} \cdot \frac{x + 8}{x + 8} - \frac{x + 4}{x + 8} = 0$

Step 2.2

To write $- \frac{x + 4}{x + 8}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{x - 9}{x + 5} \cdot \frac{x + 8}{x + 8} - \frac{x + 4}{x + 8} \cdot \frac{x + 5}{x + 5} = 0$

Step 2.3

Write each expression with a common denominator of $(x + 5) (x + 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{x - 9}{x + 5}$ by $\frac{x + 8}{x + 8}$ .

$\frac{(x - 9) (x + 8)}{(x + 5) (x + 8)} - \frac{x + 4}{x + 8} \cdot \frac{x + 5}{x + 5} = 0$

Step 2.3.2

Multiply $\frac{x + 4}{x + 8}$ by $\frac{x + 5}{x + 5}$ .

$\frac{(x - 9) (x + 8)}{(x + 5) (x + 8)} - \frac{(x + 4) (x + 5)}{(x + 8) (x + 5)} = 0$

Step 2.3.3

Reorder the factors of $(x + 8) (x + 5)$ .

$\frac{(x - 9) (x + 8)}{(x + 5) (x + 8)} - \frac{(x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{(x - 9) (x + 8) - (x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Expand $(x - 9) (x + 8)$ using the FOIL Method.

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Step 2.5.1.1

Apply the distributive property.

$\frac{x (x + 8) - 9 (x + 8) - (x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 8 - 9 (x + 8) - (x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 8 - 9 x - 9 \cdot 8 - (x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.2

Simplify and combine like terms.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 8 - 9 x - 9 \cdot 8 - (x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.2.1.2

Move $8$ to the left of $x$ .

$\frac{x^{2} + 8 \cdot x - 9 x - 9 \cdot 8 - (x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.2.1.3

Multiply $- 9$ by $8$ .

$\frac{x^{2} + 8 x - 9 x - 72 - (x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.2.2

Subtract $9 x$ from $8 x$ .

$\frac{x^{2} - x - 72 - (x + 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.3

Apply the distributive property.

$\frac{x^{2} - x - 72 + (- x - 1 \cdot 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.4

Multiply $- 1$ by $4$ .

$\frac{x^{2} - x - 72 + (- x - 4) (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.5

Expand $(- x - 4) (x + 5)$ using the FOIL Method.

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Step 2.5.5.1

Apply the distributive property.

$\frac{x^{2} - x - 72 - x (x + 5) - 4 (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.5.2

Apply the distributive property.

$\frac{x^{2} - x - 72 - x \cdot x - x \cdot 5 - 4 (x + 5)}{(x + 5) (x + 8)} = 0$

Step 2.5.5.3

Apply the distributive property.

$\frac{x^{2} - x - 72 - x \cdot x - x \cdot 5 - 4 x - 4 \cdot 5}{(x + 5) (x + 8)} = 0$

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 $x$ .

$\frac{x^{2} - x - 72 - (x \cdot x) - x \cdot 5 - 4 x - 4 \cdot 5}{(x + 5) (x + 8)} = 0$

Step 2.5.6.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} - x - 72 - x^{2} - x \cdot 5 - 4 x - 4 \cdot 5}{(x + 5) (x + 8)} = 0$

Step 2.5.6.1.2

Multiply $5$ by $- 1$ .

$\frac{x^{2} - x - 72 - x^{2} - 5 x - 4 x - 4 \cdot 5}{(x + 5) (x + 8)} = 0$

Step 2.5.6.1.3

Multiply $- 4$ by $5$ .

$\frac{x^{2} - x - 72 - x^{2} - 5 x - 4 x - 20}{(x + 5) (x + 8)} = 0$

Step 2.5.6.2

Subtract $4 x$ from $- 5 x$ .

$\frac{x^{2} - x - 72 - x^{2} - 9 x - 20}{(x + 5) (x + 8)} = 0$

Step 2.5.7

Subtract $x^{2}$ from $x^{2}$ .

$\frac{- x - 72 + 0 - 9 x - 20}{(x + 5) (x + 8)} = 0$

Step 2.5.8

Add $- x$ and $0$ .

$\frac{- x - 72 - 9 x - 20}{(x + 5) (x + 8)} = 0$

Step 2.5.9

Subtract $9 x$ from $- x$ .

$\frac{- 10 x - 72 - 20}{(x + 5) (x + 8)} = 0$

Step 2.5.10

Subtract $20$ from $- 72$ .

$\frac{- 10 x - 92}{(x + 5) (x + 8)} = 0$

Step 2.5.11

Factor $2$ out of $- 10 x - 92$ .

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Step 2.5.11.1

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

$\frac{2 (- 5 x) - 92}{(x + 5) (x + 8)} = 0$

Step 2.5.11.2

Factor $2$ out of $- 92$ .

$\frac{2 (- 5 x) + 2 (- 46)}{(x + 5) (x + 8)} = 0$

Step 2.5.11.3

Factor $2$ out of $2 (- 5 x) + 2 (- 46)$ .

$\frac{2 (- 5 x - 46)}{(x + 5) (x + 8)} = 0$

Step 2.6

Factor $- 1$ out of $- 5 x$ .

$\frac{2 (- (5 x) - 46)}{(x + 5) (x + 8)} = 0$

Step 2.7

Rewrite $- 46$ as $- 1 (46)$ .

$\frac{2 (- (5 x) - 1 (46))}{(x + 5) (x + 8)} = 0$

Step 2.8

Factor $- 1$ out of $- (5 x) - 1 (46)$ .

$\frac{2 (- (5 x + 46))}{(x + 5) (x + 8)} = 0$

Step 2.9

Rewrite $- (5 x + 46)$ as $- 1 (5 x + 46)$ .

$\frac{2 (- 1 (5 x + 46))}{(x + 5) (x + 8)} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{2 (5 x + 46)}{(x + 5) (x + 8)} = 0$

Step 3

Set the numerator equal to zero.

$2 (5 x + 46) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $2 (5 x + 46) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (5 x + 46) = 0$ by $2$ .

$\frac{2 (5 x + 46)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (5 x + 46)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $5 x + 46$ by $1$ .

$5 x + 46 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$5 x + 46 = 0$

Step 4.2

Subtract $46$ from both sides of the equation.

$5 x = - 46$

Step 4.3

Divide each term in $5 x = - 46$ by $5$ and simplify.

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Step 4.3.1

Divide each term in $5 x = - 46$ by $5$ .

$\frac{5 x}{5} = \frac{- 46}{5}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $5$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 46}{5}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 46}{5}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{46}{5}$