Solve by Factoring 4/x-5/3=x/6

$\frac{4}{x} - \frac{5}{3} = \frac{x}{6}$

Step 1

Subtract $\frac{x}{6}$ from both sides of the equation.

$\frac{4}{x} - \frac{5}{3} - \frac{x}{6} = 0$

Step 2

Reorder terms.

$- \frac{x}{6} + \frac{4}{x} - \frac{5}{3} = 0$

Step 3

To write $- \frac{x}{6}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{x}{6} \cdot \frac{x}{x} + \frac{4}{x} - \frac{5}{3} = 0$

Step 4

To write $\frac{4}{x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{x}{6} \cdot \frac{x}{x} + \frac{4}{x} \cdot \frac{6}{6} - \frac{5}{3} = 0$

Step 5

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{6}$ by $\frac{x}{x}$ .

$- \frac{x \cdot x}{6 x} + \frac{4}{x} \cdot \frac{6}{6} - \frac{5}{3} = 0$

Multiply $\frac{4}{x}$ by $\frac{6}{6}$ .

$- \frac{x \cdot x}{6 x} + \frac{4 \cdot 6}{x \cdot 6} - \frac{5}{3} = 0$

Reorder the factors of $x \cdot 6$ .

$- \frac{x \cdot x}{6 x} + \frac{4 \cdot 6}{6 x} - \frac{5}{3} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- x \cdot x + 4 \cdot 6}{6 x} - \frac{5}{3} = 0$

Step 7

Multiply $4$ by $6$ .

$\frac{- x \cdot x + 24}{6 x} - \frac{5}{3} = 0$

Step 8

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

Move $x$ .

$\frac{- (x \cdot x) + 24}{6 x} - \frac{5}{3} = 0$

Multiply $x$ by $x$ .

$\frac{- x^{2} + 24}{6 x} - \frac{5}{3} = 0$

Step 9

To write $- \frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{- x^{2} + 24}{6 x} - \frac{5}{3} \cdot \frac{2 x}{2 x} = 0$

Step 10

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{3}$ by $\frac{2 x}{2 x}$ .

$\frac{- x^{2} + 24}{6 x} - \frac{5 (2 x)}{3 (2 x)} = 0$

Multiply $2$ by $3$ .

$\frac{- x^{2} + 24}{6 x} - \frac{5 (2 x)}{6 x} = 0$

Step 11

Combine the numerators over the common denominator.

$\frac{- x^{2} + 24 - 5 (2 x)}{6 x} = 0$

Step 12

Remove parentheses.

$\frac{- x^{2} + 24 - 5 \cdot (2 x)}{6 x} = 0$

Step 13

Simplify the numerator.

Multiply $- 5$ by $2$ .

$\frac{- x^{2} + 24 - 10 x}{6 x} = 0$

Reorder terms.

$\frac{- x^{2} - 10 x + 24}{6 x} = 0$

Factor by grouping.

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 $a \cdot c = - 1 \cdot 24 = - 24$ and whose sum is $b = - 10$ .

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

$\frac{- x^{2} - 10 x + 24}{6 x} = 0$

Rewrite $- 10$ as $2$ plus $- 12$

$\frac{- x^{2} + (2 - 12) x + 24}{6 x} = 0$

Apply the distributive property.

$\frac{- x^{2} + 2 x - 12 x + 24}{6 x} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- x^{2} + 2 x) - 12 x + 24}{6 x} = 0$

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

$\frac{x (- x + 2) + 12 (- x + 2)}{6 x} = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$\frac{(- x + 2) (x + 12)}{6 x} = 0$

Step 14

Set the numerator equal to zero.

$(- x + 2) (x + 12) = 0$

Step 15

Solve the equation for $x$ .

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

$- x + 2 = 0$

$x + 12 = 0$

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

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

$- x + 2 = 0$

Solve $- x + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$- x = - 2$

Divide each term in $- x = - 2$ by $- 1$ and simplify.

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x = 2$

Set $x + 12$ equal to $0$ and solve for $x$ .

Set $x + 12$ equal to $0$ .

$x + 12 = 0$

Subtract $12$ from both sides of the equation.

$x = - 12$

The final solution is all the values that make $(- x + 2) (x + 12) = 0$ true.

$x = 2, - 12$

Solve by Factoring 4/x-5/3=x/6

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