Finite Math Examples

$\frac{5 - 3 x}{x^{2} + 4 x + 3} - \frac{2 x + 2}{x + 3} = \frac{3 - x}{x + 1}$

Step 1

Subtract $\frac{3 - x}{x + 1}$ from both sides of the equation.

$\frac{5 - 3 x}{x^{2} + 4 x + 3} - \frac{2 x + 2}{x + 3} - \frac{3 - x}{x + 1} = 0$

Step 2

Simplify $\frac{5 - 3 x}{x^{2} + 4 x + 3} - \frac{2 x + 2}{x + 3} - \frac{3 - x}{x + 1}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 2.1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 2.1.1.2

Write the factored form using these integers.

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

Step 2.1.2

Factor $2$ out of $2 x + 2$ .

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Step 2.1.2.1

Factor $2$ out of $2 x$ .

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

Step 2.1.2.2

Factor $2$ out of $2$ .

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

Step 2.1.2.3

Factor $2$ out of $2 (x) + 2 (1)$ .

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

Step 2.2

Find the common denominator.

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Step 2.2.1

Multiply $\frac{2 (x + 1)}{x + 3}$ by $\frac{x + 1}{x + 1}$ .

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

Step 2.2.2

Multiply $\frac{2 (x + 1)}{x + 3}$ by $\frac{x + 1}{x + 1}$ .

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

Step 2.2.3

Multiply $\frac{3 - x}{x + 1}$ by $\frac{x + 3}{x + 3}$ .

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

Step 2.2.4

Multiply $\frac{3 - x}{x + 1}$ by $\frac{x + 3}{x + 3}$ .

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

Step 2.2.5

Reorder the factors of $(x + 3) (x + 1)$ .

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify each term.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Multiply $- 2$ by $1$ .

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

Step 2.4.3

Expand $(- 2 x - 2) (x + 1)$ using the FOIL Method.

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Step 2.4.3.1

Apply the distributive property.

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

Step 2.4.3.2

Apply the distributive property.

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

Step 2.4.3.3

Apply the distributive property.

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

Step 2.4.4

Simplify and combine like terms.

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Step 2.4.4.1

Simplify each term.

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Step 2.4.4.1.1

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

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Step 2.4.4.1.1.1

Move $x$ .

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

Step 2.4.4.1.1.2

Multiply $x$ by $x$ .

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

Step 2.4.4.1.2

Multiply $- 2$ by $1$ .

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

Step 2.4.4.1.3

Multiply $- 2$ by $1$ .

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

Step 2.4.4.2

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

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 - (3 - x) (x + 3)}{(x + 1) (x + 3)} = 0$

Step 2.4.5

Apply the distributive property.

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 + (- 1 \cdot 3 - - x) (x + 3)}{(x + 1) (x + 3)} = 0$

Step 2.4.6

Multiply $- 1$ by $3$ .

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 + (- 3 - - x) (x + 3)}{(x + 1) (x + 3)} = 0$

Step 2.4.7

Multiply $- - x$ .

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Step 2.4.7.1

Multiply $- 1$ by $- 1$ .

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 + (- 3 + 1 x) (x + 3)}{(x + 1) (x + 3)} = 0$

Step 2.4.7.2

Multiply $x$ by $1$ .

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 + (- 3 + x) (x + 3)}{(x + 1) (x + 3)} = 0$

Step 2.4.8

Expand $(- 3 + x) (x + 3)$ using the FOIL Method.

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Step 2.4.8.1

Apply the distributive property.

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 - 3 (x + 3) + x (x + 3)}{(x + 1) (x + 3)} = 0$

Step 2.4.8.2

Apply the distributive property.

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 - 3 x - 3 \cdot 3 + x (x + 3)}{(x + 1) (x + 3)} = 0$

Step 2.4.8.3

Apply the distributive property.

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 - 3 x - 3 \cdot 3 + x \cdot x + x \cdot 3}{(x + 1) (x + 3)} = 0$

Step 2.4.9

Combine the opposite terms in $- 3 x - 3 \cdot 3 + x \cdot x + x \cdot 3$ .

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Step 2.4.9.1

Reorder the factors in the terms $- 3 x$ and $x \cdot 3$ .

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 - 3 x - 3 \cdot 3 + x \cdot x + 3 x}{(x + 1) (x + 3)} = 0$

Step 2.4.9.2

Add $- 3 x$ and $3 x$ .

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 + 0 - 3 \cdot 3 + x \cdot x}{(x + 1) (x + 3)} = 0$

Step 2.4.9.3

Subtract $3 \cdot 3$ from $0$ .

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 - 3 \cdot 3 + x \cdot x}{(x + 1) (x + 3)} = 0$

Step 2.4.10

Simplify each term.

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Step 2.4.10.1

Multiply $- 3$ by $3$ .

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 - 9 + x \cdot x}{(x + 1) (x + 3)} = 0$

Step 2.4.10.2

Multiply $x$ by $x$ .

$\frac{5 - 3 x - 2 x^{2} - 4 x - 2 - 9 + x^{2}}{(x + 1) (x + 3)} = 0$

Step 2.5

Subtract $2$ from $5$ .

$\frac{- 3 x - 2 x^{2} - 4 x + 3 - 9 + x^{2}}{(x + 1) (x + 3)} = 0$

Step 2.6

Subtract $4 x$ from $- 3 x$ .

$\frac{- 2 x^{2} - 7 x + 3 - 9 + x^{2}}{(x + 1) (x + 3)} = 0$

Step 2.7

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

$\frac{- x^{2} - 7 x + 3 - 9}{(x + 1) (x + 3)} = 0$

Step 2.8

Subtract $9$ from $3$ .

$\frac{- x^{2} - 7 x - 6}{(x + 1) (x + 3)} = 0$

Step 2.9

Factor by grouping.

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Step 2.9.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 6 = 6$ and whose sum is $b = - 7$ .

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Step 2.9.1.1

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

$\frac{- x^{2} - 7 (x) - 6}{(x + 1) (x + 3)} = 0$

Step 2.9.1.2

Rewrite $- 7$ as $- 1$ plus $- 6$

$\frac{- x^{2} + (- 1 - 6) x - 6}{(x + 1) (x + 3)} = 0$

Step 2.9.1.3

Apply the distributive property.

$\frac{- x^{2} - 1 x - 6 x - 6}{(x + 1) (x + 3)} = 0$

Step 2.9.2

Factor out the greatest common factor from each group.

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Step 2.9.2.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 1 x) - 6 x - 6}{(x + 1) (x + 3)} = 0$

Step 2.9.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x - 1) + 6 (- x - 1)}{(x + 1) (x + 3)} = 0$

Step 2.9.3

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$\frac{(- x - 1) (x + 6)}{(x + 1) (x + 3)} = 0$

Step 2.10

Cancel the common factor of $- x - 1$ and $x + 1$ .

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Step 2.10.1

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

$\frac{(- (x) - 1) (x + 6)}{(x + 1) (x + 3)} = 0$

Step 2.10.2

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

$\frac{(- (x) - 1 (1)) (x + 6)}{(x + 1) (x + 3)} = 0$

Step 2.10.3

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

$\frac{- (x + 1) (x + 6)}{(x + 1) (x + 3)} = 0$

Step 2.10.4

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

$\frac{- 1 (x + 1) (x + 6)}{(x + 1) (x + 3)} = 0$

Step 2.10.5

Cancel the common factor.

$\frac{- 1 (x + 1) (x + 6)}{(x + 1) (x + 3)} = 0$

Step 2.10.6

Rewrite the expression.

$\frac{- 1 (x + 6)}{x + 3} = 0$

Step 2.11

Move the negative in front of the fraction.

$- \frac{x + 6}{x + 3} = 0$