Finite Math Examples

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

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

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Step 2.1

To write $\frac{x}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 2.2

To write $- \frac{1}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Step 2.3

Write each expression with a common denominator of $(x + 4) (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{x}{x + 4}$ by $\frac{x + 1}{x + 1}$ .

$\frac{x (x + 1)}{(x + 4) (x + 1)} - \frac{1}{x + 1} \cdot \frac{x + 4}{x + 4} - 1 = 0$

Step 2.3.2

Multiply $\frac{1}{x + 1}$ by $\frac{x + 4}{x + 4}$ .

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

Step 2.3.3

Reorder the factors of $(x + 4) (x + 1)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 - (x + 4)}{(x + 1) (x + 4)} - 1 = 0$

Step 2.5.2

Multiply $x$ by $x$ .

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

Step 2.5.3

Multiply $x$ by $1$ .

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

Step 2.5.4

Apply the distributive property.

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

Step 2.5.5

Multiply $- 1$ by $4$ .

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

Step 2.5.6

Subtract $x$ from $x$ .

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

Step 2.5.7

Add $x^{2}$ and $0$ .

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

Step 2.5.8

Rewrite $x^{2} - 4$ in a factored form.

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Step 2.5.8.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.5.8.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 2.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{(x + 1) (x + 4)}{(x + 1) (x + 4)}$ .

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

Step 2.7

Combine $- 1$ and $\frac{(x + 1) (x + 4)}{(x + 1) (x + 4)}$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

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

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Step 2.9.1.1

Apply the distributive property.

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

Step 2.9.1.2

Apply the distributive property.

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

Step 2.9.1.3

Apply the distributive property.

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

Step 2.9.2

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 2.9.2.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

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

Step 2.9.2.2

Add $- 2 x$ and $2 x$ .

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

Step 2.9.2.3

Add $x \cdot x$ and $0$ .

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

Step 2.9.3

Simplify each term.

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Step 2.9.3.1

Multiply $x$ by $x$ .

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

Step 2.9.3.2

Multiply $2$ by $- 2$ .

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

Step 2.9.4

Apply the distributive property.

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

Step 2.9.5

Multiply $- 1$ by $1$ .

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

Step 2.9.6

Expand $(- x - 1) (x + 4)$ using the FOIL Method.

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Step 2.9.6.1

Apply the distributive property.

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

Step 2.9.6.2

Apply the distributive property.

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

Step 2.9.6.3

Apply the distributive property.

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

Step 2.9.7

Simplify and combine like terms.

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Step 2.9.7.1

Simplify each term.

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Step 2.9.7.1.1

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

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Step 2.9.7.1.1.1

Move $x$ .

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

Step 2.9.7.1.1.2

Multiply $x$ by $x$ .

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

Step 2.9.7.1.2

Multiply $4$ by $- 1$ .

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

Step 2.9.7.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.9.7.1.4

Multiply $- 1$ by $4$ .

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

Step 2.9.7.2

Subtract $x$ from $- 4 x$ .

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

Step 2.9.8

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

$\frac{0 - 4 - 5 x - 4}{(x + 1) (x + 4)} = 0$

Step 2.9.9

Subtract $4$ from $0$ .

$\frac{- 4 - 5 x - 4}{(x + 1) (x + 4)} = 0$

Step 2.9.10

Subtract $4$ from $- 4$ .

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

Step 2.10

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

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

Step 2.11

Rewrite $- 8$ as $- 1 (8)$ .

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

Step 2.12

Factor $- 1$ out of $- (5 x) - 1 (8)$ .

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

Step 2.13

Rewrite $- (5 x + 8)$ as $- 1 (5 x + 8)$ .

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

Step 2.14

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$5 x + 8 = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Subtract $8$ from both sides of the equation.

$5 x = - 8$

Step 4.2

Divide each term in $5 x = - 8$ by $5$ and simplify.

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Step 4.2.1

Divide each term in $5 x = - 8$ by $5$ .

$\frac{5 x}{5} = \frac{- 8}{5}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 8}{5}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 8}{5}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move the negative in front of the fraction.

$x = - \frac{8}{5}$