Finite Math Examples

$\log_{3} (5 - x) + \log_{3} (x + 1) = \log_{3} (3 x - 1)$

Step 1

Subtract $\log_{3} (3 x - 1)$ from both sides of the equation.

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

Step 2

Simplify $\log_{3} (5 - x) + \log_{3} (x + 1) - \log_{3} (3 x - 1)$ .

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Step 2.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 2.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 3

Rewrite $\log_{3} (\frac{(5 - x) (x + 1)}{3 x - 1}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

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

Step 4

Cross multiply to remove the fraction.

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

Step 5

Simplify $3^{0} (3 x - 1)$ .

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Step 5.1

Anything raised to $0$ is $1$ .

$(5 - x) (x + 1) = 1 (3 x - 1)$

Step 5.2

Multiply $3 x - 1$ by $1$ .

$(5 - x) (x + 1) = 3 x - 1$

Step 6

Move all terms containing $x$ to the left side of the equation.

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Step 6.1

Subtract $3 x$ from both sides of the equation.

$(5 - x) (x + 1) - 3 x = - 1$

Step 6.2

Simplify each term.

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Step 6.2.1

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

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Step 6.2.1.1

Apply the distributive property.

$5 (x + 1) - x (x + 1) - 3 x = - 1$

Step 6.2.1.2

Apply the distributive property.

$5 x + 5 \cdot 1 - x (x + 1) - 3 x = - 1$

Step 6.2.1.3

Apply the distributive property.

$5 x + 5 \cdot 1 - x \cdot x - x \cdot 1 - 3 x = - 1$

Step 6.2.2

Simplify and combine like terms.

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Step 6.2.2.1

Simplify each term.

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Step 6.2.2.1.1

Multiply $5$ by $1$ .

$5 x + 5 - x \cdot x - x \cdot 1 - 3 x = - 1$

Step 6.2.2.1.2

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

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Step 6.2.2.1.2.1

Move $x$ .

$5 x + 5 - (x \cdot x) - x \cdot 1 - 3 x = - 1$

Step 6.2.2.1.2.2

Multiply $x$ by $x$ .

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

Step 6.2.2.1.3

Multiply $- 1$ by $1$ .

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

Step 6.2.2.2

Subtract $x$ from $5 x$ .

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

Step 6.3

Subtract $3 x$ from $4 x$ .

$x + 5 - x^{2} = - 1$

Step 7

Move all terms not containing $x$ to the right side of the equation.

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Step 7.1

Subtract $5$ from both sides of the equation.

$x - x^{2} = - 1 - 5$

Step 7.2

Subtract $5$ from $- 1$ .

$x - x^{2} = - 6$

Step 8

Add $6$ to both sides of the equation.

$x - x^{2} + 6 = 0$

Step 9

Factor the left side of the equation.

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Step 9.1

Factor $- 1$ out of $x - x^{2} + 6$ .

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Step 9.1.1

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

$- x^{2} + x + 6 = 0$

Step 9.1.2

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

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

Step 9.1.3

Factor $- 1$ out of $x$ .

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

Step 9.1.4

Rewrite $6$ as $- 1 (- 6)$ .

$- (x^{2}) - 1 (- x) - 1 \cdot - 6 = 0$

Step 9.1.5

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

$- (x^{2} - x) - 1 \cdot - 6 = 0$

Step 9.1.6

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 6)$ .

$- (x^{2} - x - 6) = 0$

Step 9.2

Factor.

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Step 9.2.1

Factor $x^{2} - x - 6$ using the AC method.

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Step 9.2.1.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 9.2.1.2

Write the factored form using these integers.

$- ((x - 3) (x + 2)) = 0$

Step 9.2.2

Remove unnecessary parentheses.

$- (x - 3) (x + 2) = 0$

Step 10

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 2 = 0$

Step 11

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 11.2

Add $3$ to both sides of the equation.

$x = 3$

Step 12

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

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Step 12.1

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

$x + 2 = 0$

Step 12.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 13

The final solution is all the values that make $- (x - 3) (x + 2) = 0$ true.

$x = 3, - 2$