Finite Math Examples

$\log_{2} (3 x + 1) - \log_{2} (x + 2) + 2 = \log_{2} (9 x - 4) - \log_{2} (x)$

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 2

Simplify the right side.

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Step 2.1

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

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

Step 3

Move all the terms containing a logarithm to the left side of the equation.

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

Step 4

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

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Multiply $\frac{3 x + 1}{x + 2}$ by $\frac{x}{9 x - 4}$ .

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

Step 7

Reorder factors in $\log_{2} (\frac{(3 x + 1) x}{(x + 2) (9 x - 4)})$ .

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

Step 8

Rewrite $\log_{2} (\frac{x (3 x + 1)}{(x + 2) (9 x - 4)}) = - 2$ 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 $b^{y} = x$ .

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

Step 9

Cross multiply to remove the fraction.

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

Step 10

Simplify $2^{- 2} ((x + 2) (9 x - 4))$ .

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Step 10.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10.2

Raise $2$ to the power of $2$ .

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

Step 10.3

Expand $(x + 2) (9 x - 4)$ using the FOIL Method.

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Step 10.3.1

Apply the distributive property.

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

Step 10.3.2

Apply the distributive property.

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

Step 10.3.3

Apply the distributive property.

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

Step 10.4

Simplify and combine like terms.

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Step 10.4.1

Simplify each term.

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Step 10.4.1.1

Rewrite using the commutative property of multiplication.

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

Step 10.4.1.2

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

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Step 10.4.1.2.1

Move $x$ .

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

Step 10.4.1.2.2

Multiply $x$ by $x$ .

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

Step 10.4.1.3

Move $- 4$ to the left of $x$ .

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

Step 10.4.1.4

Multiply $9$ by $2$ .

$x (3 x + 1) = \frac{1}{4} (9 x^{2} - 4 x + 18 x + 2 \cdot - 4)$

Step 10.4.1.5

Multiply $2$ by $- 4$ .

$x (3 x + 1) = \frac{1}{4} (9 x^{2} - 4 x + 18 x - 8)$

Step 10.4.2

Add $- 4 x$ and $18 x$ .

$x (3 x + 1) = \frac{1}{4} (9 x^{2} + 14 x - 8)$

Step 10.5

Apply the distributive property.

$x (3 x + 1) = \frac{1}{4} (9 x^{2}) + \frac{1}{4} (14 x) + \frac{1}{4} \cdot - 8$

Step 10.6

Simplify.

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Step 10.6.1

Multiply $\frac{1}{4} (9 x^{2})$ .

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Step 10.6.1.1

Combine $9$ and $\frac{1}{4}$ .

$x (3 x + 1) = \frac{9}{4} x^{2} + \frac{1}{4} (14 x) + \frac{1}{4} \cdot - 8$

Step 10.6.1.2

Combine $\frac{9}{4}$ and $x^{2}$ .

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{1}{4} (14 x) + \frac{1}{4} \cdot - 8$

Step 10.6.2

Cancel the common factor of $2$ .

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Step 10.6.2.1

Factor $2$ out of $4$ .

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{1}{2 (2)} (14 x) + \frac{1}{4} \cdot - 8$

Step 10.6.2.2

Factor $2$ out of $14 x$ .

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{1}{2 (2)} (2 (7 x)) + \frac{1}{4} \cdot - 8$

Step 10.6.2.3

Cancel the common factor.

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{1}{2 \cdot 2} (2 (7 x)) + \frac{1}{4} \cdot - 8$

Step 10.6.2.4

Rewrite the expression.

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{1}{2} (7 x) + \frac{1}{4} \cdot - 8$

Step 10.6.3

Combine $7$ and $\frac{1}{2}$ .

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{7}{2} x + \frac{1}{4} \cdot - 8$

Step 10.6.4

Combine $\frac{7}{2}$ and $x$ .

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{7 x}{2} + \frac{1}{4} \cdot - 8$

Step 10.6.5

Cancel the common factor of $4$ .

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Step 10.6.5.1

Factor $4$ out of $- 8$ .

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{7 x}{2} + \frac{1}{4} \cdot (4 (- 2))$

Step 10.6.5.2

Cancel the common factor.

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{7 x}{2} + \frac{1}{4} \cdot (4 \cdot - 2)$

Step 10.6.5.3

Rewrite the expression.

$x (3 x + 1) = \frac{9 x^{2}}{4} + \frac{7 x}{2} - 2$

Step 11

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

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Step 11.1

Subtract $\frac{9 x^{2}}{4}$ from both sides of the equation.

$x (3 x + 1) - \frac{9 x^{2}}{4} = \frac{7 x}{2} - 2$

Step 11.2

Subtract $\frac{7 x}{2}$ from both sides of the equation.

$x (3 x + 1) - \frac{9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.3

Simplify each term.

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Step 11.3.1

Apply the distributive property.

$x (3 x) + x \cdot 1 - \frac{9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.3.2

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot 1 - \frac{9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.3.3

Multiply $x$ by $1$ .

$3 x \cdot x + x - \frac{9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.3.4

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

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Step 11.3.4.1

Move $x$ .

$3 (x \cdot x) + x - \frac{9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.3.4.2

Multiply $x$ by $x$ .

$3 x^{2} + x - \frac{9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.4

To write $3 x^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x + 3 x^{2} \cdot \frac{4}{4} - \frac{9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.5

Combine $3 x^{2}$ and $\frac{4}{4}$ .

$x + \frac{3 x^{2} \cdot 4}{4} - \frac{9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.6

Combine the numerators over the common denominator.

$x + \frac{3 x^{2} \cdot 4 - 9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.7

Find the common denominator.

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Step 11.7.1

Write $x$ as a fraction with denominator $1$ .

$\frac{x}{1} + \frac{3 x^{2} \cdot 4 - 9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.7.2

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

$\frac{x}{1} \cdot \frac{4}{4} + \frac{3 x^{2} \cdot 4 - 9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.7.3

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

$\frac{x \cdot 4}{4} + \frac{3 x^{2} \cdot 4 - 9 x^{2}}{4} - \frac{7 x}{2} = - 2$

Step 11.7.4

Multiply $\frac{7 x}{2}$ by $\frac{2}{2}$ .

$\frac{x \cdot 4}{4} + \frac{3 x^{2} \cdot 4 - 9 x^{2}}{4} - (\frac{7 x}{2} \cdot \frac{2}{2}) = - 2$

Step 11.7.5

Multiply $\frac{7 x}{2}$ by $\frac{2}{2}$ .

$\frac{x \cdot 4}{4} + \frac{3 x^{2} \cdot 4 - 9 x^{2}}{4} - \frac{7 x \cdot 2}{2 \cdot 2} = - 2$

Step 11.7.6

Multiply $2$ by $2$ .

$\frac{x \cdot 4}{4} + \frac{3 x^{2} \cdot 4 - 9 x^{2}}{4} - \frac{7 x \cdot 2}{4} = - 2$

Step 11.8

Combine the numerators over the common denominator.

$\frac{x \cdot 4 + 3 x^{2} \cdot 4 - 9 x^{2} - 7 x \cdot 2}{4} = - 2$

Step 11.9

Simplify each term.

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Step 11.9.1

Move $4$ to the left of $x$ .

$\frac{4 \cdot x + 3 x^{2} \cdot 4 - 9 x^{2} - 7 x \cdot 2}{4} = - 2$

Step 11.9.2

Multiply $4$ by $3$ .

$\frac{4 x + 12 x^{2} - 9 x^{2} - 7 x \cdot 2}{4} = - 2$

Step 11.9.3

Multiply $2$ by $- 7$ .

$\frac{4 x + 12 x^{2} - 9 x^{2} - 14 x}{4} = - 2$

Step 11.10

Subtract $14 x$ from $4 x$ .

$\frac{12 x^{2} - 9 x^{2} - 10 x}{4} = - 2$

Step 11.11

Subtract $9 x^{2}$ from $12 x^{2}$ .

$\frac{3 x^{2} - 10 x}{4} = - 2$

Step 11.12

Factor $x$ out of $3 x^{2} - 10 x$ .

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Step 11.12.1

Factor $x$ out of $3 x^{2}$ .

$\frac{x (3 x) - 10 x}{4} = - 2$

Step 11.12.2

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

$\frac{x (3 x) + x \cdot - 10}{4} = - 2$

Step 11.12.3

Factor $x$ out of $x (3 x) + x \cdot - 10$ .

$\frac{x (3 x - 10)}{4} = - 2$

Step 12

Multiply both sides by $4$ .

$\frac{x (3 x - 10)}{4} \cdot 4 = - 2 \cdot 4$

Step 13

Simplify.

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Step 13.1

Simplify the left side.

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Step 13.1.1

Simplify $\frac{x (3 x - 10)}{4} \cdot 4$ .

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Step 13.1.1.1

Simplify terms.

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Step 13.1.1.1.1

Cancel the common factor of $4$ .

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Step 13.1.1.1.1.1

Cancel the common factor.

$\frac{x (3 x - 10)}{4} \cdot 4 = - 2 \cdot 4$

Step 13.1.1.1.1.2

Rewrite the expression.

$x (3 x - 10) = - 2 \cdot 4$

Step 13.1.1.1.2

Apply the distributive property.

$x (3 x) + x \cdot - 10 = - 2 \cdot 4$

Step 13.1.1.1.3

Reorder.

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Step 13.1.1.1.3.1

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot - 10 = - 2 \cdot 4$

Step 13.1.1.1.3.2

Move $- 10$ to the left of $x$ .

$3 x \cdot x - 10 \cdot x = - 2 \cdot 4$

Step 13.1.1.2

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

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Step 13.1.1.2.1

Move $x$ .

$3 (x \cdot x) - 10 \cdot x = - 2 \cdot 4$

Step 13.1.1.2.2

Multiply $x$ by $x$ .

$3 x^{2} - 10 \cdot x = - 2 \cdot 4$

$3 x^{2} - 10 x = - 2 \cdot 4$

Step 13.2

Simplify the right side.

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Step 13.2.1

Multiply $- 2$ by $4$ .

$3 x^{2} - 10 x = - 8$

Step 14

Solve for $x$ .

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Step 14.1

Add $8$ to both sides of the equation.

$3 x^{2} - 10 x + 8 = 0$

Step 14.2

Factor by grouping.

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Step 14.2.1

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 $a \cdot c = 3 \cdot 8 = 24$ and whose sum is $b = - 10$ .

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Step 14.2.1.1

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

$3 x^{2} - 10 x + 8 = 0$

Step 14.2.1.2

Rewrite $- 10$ as $- 4$ plus $- 6$

$3 x^{2} + (- 4 - 6) x + 8 = 0$

Step 14.2.1.3

Apply the distributive property.

$3 x^{2} - 4 x - 6 x + 8 = 0$

Step 14.2.2

Factor out the greatest common factor from each group.

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Step 14.2.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 4 x) - 6 x + 8 = 0$

Step 14.2.2.2

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

$x (3 x - 4) - 2 (3 x - 4) = 0$

Step 14.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 4$ .

$(3 x - 4) (x - 2) = 0$

Step 14.3

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

$3 x - 4 = 0$

$x - 2 = 0$

Step 14.4

Set $3 x - 4$ equal to $0$ and solve for $x$ .

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Step 14.4.1

Set $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 14.4.2

Solve $3 x - 4 = 0$ for $x$ .

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Step 14.4.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 14.4.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 14.4.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 14.4.2.2.2

Simplify the left side.

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Step 14.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 14.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 14.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 14.5

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

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Step 14.5.1

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

$x - 2 = 0$

Step 14.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 14.6

The final solution is all the values that make $(3 x - 4) (x - 2) = 0$ true.

$x = \frac{4}{3}, 2$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3}, 2$

Decimal Form:

$x = 1.\overline{3}, 2$

Mixed Number Form:

$x = 1 \frac{1}{3}, 2$