Solve by Factoring 3/(2x)-1/(2(x+8))=1

$\frac{3}{2 x} - \frac{1}{2 (x + 8)} = 1$

Move $1$ to the left side of the equation by subtracting it from both sides.

$\frac{3}{2 x} - \frac{1}{2 (x + 8)} - 1 = 0$

Reorder terms.

$\frac{3}{2 x} - 1 - \frac{1}{2 (x + 8)} = 0$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{3}{2 x} - 1 \cdot \frac{2 x}{2 x} - \frac{1}{2 (x + 8)} = 0$

Combine $- 1$ and $\frac{2 x}{2 x}$ .

$\frac{3}{2 x} + \frac{- (2 x)}{2 x} - \frac{1}{2 (x + 8)} = 0$

Combine the numerators over the common denominator.

$\frac{3 - (2 x)}{2 x} - \frac{1}{2 (x + 8)} = 0$

To write $\frac{3 - (2 x)}{2 x}$ as a fraction with a common denominator, multiply by $\frac{x + 8}{x + 8}$ .

$\frac{3 - (2 x)}{2 x} \cdot \frac{x + 8}{x + 8} - \frac{1}{2 (x + 8)} = 0$

To write $- \frac{1}{2 (x + 8)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3 - (2 x)}{2 x} \cdot \frac{x + 8}{x + 8} - \frac{1}{2 (x + 8)} \cdot \frac{x}{x} = 0$

Write each expression with a common denominator of $2 x (x + 8)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{3 - (2 x)}{2 x}$ and $\frac{x + 8}{x + 8}$ .

$\frac{(3 - (2 x)) (x + 8)}{2 x (x + 8)} - \frac{1}{2 (x + 8)} \cdot \frac{x}{x} = 0$

Multiply $\frac{1}{2 (x + 8)}$ and $\frac{x}{x}$ .

$\frac{(3 - (2 x)) (x + 8)}{2 x (x + 8)} - \frac{x}{2 (x + 8) x} = 0$

Reorder the factors of $2 (x + 8) x$ .

$\frac{(3 - (2 x)) (x + 8)}{2 x (x + 8)} - \frac{x}{2 x (x + 8)} = 0$

Combine the numerators over the common denominator.

$\frac{(3 - (2 x)) (x + 8) - x}{2 x (x + 8)} = 0$

Rewrite $\frac{(3 - (2 x)) (x + 8) - x}{2 x (x + 8)}$ in a factored form.

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Multiply $2$ by $- 1$ .

$\frac{(3 - 2 x) (x + 8) - x}{2 x (x + 8)} = 0$

Expand $(3 - 2 x) (x + 8)$ using the FOIL Method.

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Apply the distributive property.

$\frac{3 (x + 8) - 2 x (x + 8) - x}{2 x (x + 8)} = 0$

Apply the distributive property.

$\frac{3 x + 3 \cdot 8 - 2 x (x + 8) - x}{2 x (x + 8)} = 0$

Apply the distributive property.

$\frac{3 x + 3 \cdot 8 - 2 x \cdot x - 2 x \cdot 8 - x}{2 x (x + 8)} = 0$

Simplify and combine like terms.

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Simplify each term.

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Multiply $3$ by $8$ .

$\frac{3 x + 24 - 2 x \cdot x - 2 x \cdot 8 - x}{2 x (x + 8)} = 0$

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

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Move $x$ .

$\frac{3 x + 24 - 2 (x \cdot x) - 2 x \cdot 8 - x}{2 x (x + 8)} = 0$

Multiply $x$ by $x$ .

$\frac{3 x + 24 - 2 x^{2} - 2 x \cdot 8 - x}{2 x (x + 8)} = 0$

Multiply $8$ by $- 2$ .

$\frac{3 x + 24 - 2 x^{2} - 16 x - x}{2 x (x + 8)} = 0$

Subtract $16 x$ from $3 x$ .

$\frac{- 13 x + 24 - 2 x^{2} - x}{2 x (x + 8)} = 0$

Subtract $x$ from $- 13 x$ .

$\frac{- 14 x + 24 - 2 x^{2}}{2 x (x + 8)} = 0$

Reorder terms.

$\frac{- 2 x^{2} - 14 x + 24}{2 x (x + 8)} = 0$

Factor $2$ out of $- 2 x^{2} - 14 x + 24$ .

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

$\frac{2 (- x^{2}) - 14 x + 24}{2 x (x + 8)} = 0$

Factor $2$ out of $- 14 x$ .

$\frac{2 (- x^{2}) + 2 (- 7 x) + 24}{2 x (x + 8)} = 0$

Factor $2$ out of $24$ .

$\frac{2 (- x^{2}) + 2 (- 7 x) + 2 (12)}{2 x (x + 8)} = 0$

Factor $2$ out of $2 (- x^{2}) + 2 (- 7 x)$ .

$\frac{2 (- x^{2} - 7 x) + 2 (12)}{2 x (x + 8)} = 0$

Factor $2$ out of $2 (- x^{2} - 7 x) + 2 (12)$ .

$\frac{2 (- x^{2} - 7 x + 12)}{2 x (x + 8)} = 0$

Reduce the expression $\frac{2 (- x^{2} - 7 x + 12)}{2 x (x + 8)}$ by cancelling the common factors.

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Cancel the common factor.

$\frac{2 (- x^{2} - 7 x + 12)}{2 x (x + 8)} = 0$

Rewrite the expression.

$\frac{- x^{2} - 7 x + 12}{x (x + 8)} = 0$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x (x + 8), 1$

Since $x (x + 8), 1$ contain both numbers and variables, there are four steps to find the LCM. Find LCM for the numeric, variable, and compound variable parts. Then, multiply them all together.

Steps to find the LCM for $x (x + 8), 1$ are:

1. Find the LCM for the numeric part $1, 1$ .

2. Find the LCM for the variable part $x^{1}$ .

3. Find the LCM for the compound variable part $x + 8$ .

4. Multiply each LCM together.

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

The LCM of $x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

The factor for $x + 8$ is $x + 8$ itself.

$(x + 8) = x + 8$

$(x + 8)$ occurs $1$ time.

The LCM of $x + 8$ is the result of multiplying all factors the greatest number of times they occur in either term.

$x + 8$

The Least Common Multiple $LCM$ of some numbers is the smallest number that the numbers are factors of.

$x (x + 8)$

Multiply each term by $x (x + 8)$ and simplify.

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Multiply each term in $\frac{- x^{2} - 7 x + 12}{x (x + 8)} = 0$ by $x (x + 8)$ in order to remove all the denominators from the equation.

$\frac{- x^{2} - 7 x + 12}{x (x + 8)} \cdot (x (x + 8)) = 0 \cdot (x (x + 8))$

Cancel the common factor of $x (x + 8)$ .

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Cancel the common factor.

$\frac{- x^{2} - 7 x + 12}{x (x + 8)} \cdot (x (x + 8)) = 0 \cdot (x (x + 8))$

Rewrite the expression.

$- x^{2} - 7 x + 12 = 0 \cdot (x (x + 8))$

Simplify $0 \cdot (x (x + 8))$ .

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Apply the distributive property.

$- x^{2} - 7 x + 12 = 0 \cdot (x \cdot x + x \cdot 8)$

Simplify the expression.

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Multiply $x$ by $x$ .

$- x^{2} - 7 x + 12 = 0 \cdot (x^{2} + x \cdot 8)$

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

$- x^{2} - 7 x + 12 = 0 \cdot (x^{2} + 8 \cdot x)$

Multiply $0$ by $x^{2} + 8 x$ .

$- x^{2} - 7 x + 12 = 0$

Solve the equation.

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Factor $- 1$ out of $- x^{2} - 7 x + 12$ .

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

$- (x^{2}) - 7 x + 12 = 0$

Factor $- 1$ out of $- 7 x$ .

$- (x^{2}) - (7 x) + 12 = 0$

Rewrite $12$ as $- 1 (- 12)$ .

$- (x^{2}) - (7 x) - 1 \cdot - 12 = 0$

Factor $- 1$ out of $- (x^{2}) - (7 x)$ .

$- (x^{2} + 7 x) - 1 \cdot - 12 = 0$

Factor $- 1$ out of $- (x^{2} + 7 x) - 1 (- 12)$ .

$- (x^{2} + 7 x - 12) = 0$

Multiply each term in $- (x^{2} + 7 x - 12) = 0$ by $- 1$

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Multiply each term in $- (x^{2} + 7 x - 12) = 0$ by $- 1$ .

$- (x^{2} + 7 x - 12) \cdot - 1 = 0 \cdot - 1$

Simplify $- (x^{2} + 7 x - 12) \cdot - 1$ .

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Apply the distributive property.

$(- x^{2} - (7 x) - - 12) \cdot - 1 = 0 \cdot - 1$

Simplify.

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Multiply $7$ by $- 1$ .

$(- x^{2} - 7 x - - 12) \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $- 12$ .

$(- x^{2} - 7 x + 12) \cdot - 1 = 0 \cdot - 1$

Apply the distributive property.

$- x^{2} \cdot - 1 - 7 x \cdot - 1 + 12 \cdot - 1 = 0 \cdot - 1$

Simplify.

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Multiply $- x^{2} \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 x^{2} - 7 x \cdot - 1 + 12 \cdot - 1 = 0 \cdot - 1$

Multiply $x^{2}$ by $1$ .

$x^{2} - 7 x \cdot - 1 + 12 \cdot - 1 = 0 \cdot - 1$

Multiply $- 1$ by $- 7$ .

$x^{2} + 7 x + 12 \cdot - 1 = 0 \cdot - 1$

Multiply $12$ by $- 1$ .

$x^{2} + 7 x - 12 = 0 \cdot - 1$

Multiply $0$ by $- 1$ .

$x^{2} + 7 x - 12 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 7$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Simplify.

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Simplify the numerator.

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Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Multiply $- 12$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{- 7 \pm \sqrt{49 + 48}}{2 \cdot 1}$

Add $49$ and $48$ .

$x = \frac{- 7 \pm \sqrt{97}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{97}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Multiply $- 12$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{- 7 \pm \sqrt{49 + 48}}{2 \cdot 1}$

Add $49$ and $48$ .

$x = \frac{- 7 \pm \sqrt{97}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{97}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{97}}{2}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{97}}{2}$

Factor $- 1$ out of $\sqrt{97}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{97})}{2}$

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{97})$ .

$x = \frac{- 1 (7 - \sqrt{97})}{2}$

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{97}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Multiply $- 12$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{- 7 \pm \sqrt{49 + 48}}{2 \cdot 1}$

Add $49$ and $48$ .

$x = \frac{- 7 \pm \sqrt{97}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{97}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{97}}{2}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{97}}{2}$

Factor $- 1$ out of $- \sqrt{97}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{97})}{2}$

Factor $- 1$ out of $- 1 (7) - (\sqrt{97})$ .

$x = \frac{- 1 (7 + \sqrt{97})}{2}$

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{97}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{97}}{2}, - \frac{7 + \sqrt{97}}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7 - \sqrt{97}}{2}, - \frac{7 + \sqrt{97}}{2}$

Decimal Form:

$x = 1.42442890 \dots, - 8.42442890 \dots$

Solve by Factoring 3/(2x)-1/(2(x+8))=1

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