Solve by Factoring 6x-5=6/x

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

Step 1

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

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

Step 2

Reorder terms.

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

Step 3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Factor $6$ out of $6 x \cdot x - 6$ .

Factor $6$ out of $6 x \cdot x$ .

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

Factor $6$ out of $- 6$ .

$\frac{6 (x \cdot x) + 6 (- 1)}{x} - 5 = 0$

Factor $6$ out of $6 (x \cdot x) + 6 (- 1)$ .

$\frac{6 (x \cdot x - 1)}{x} - 5 = 0$

Raise $x$ to the power of $1$ .

$\frac{6 (x \cdot x - 1)}{x} - 5 = 0$

Raise $x$ to the power of $1$ .

$\frac{6 (x \cdot x - 1)}{x} - 5 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 (x^{1 + 1} - 1)}{x} - 5 = 0$

Add $1$ and $1$ .

$\frac{6 (x^{2} - 1)}{x} - 5 = 0$

Rewrite $1$ as $1^{2}$ .

$\frac{6 (x^{2} - 1^{2})}{x} - 5 = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{6 ((x + 1) (x - 1))}{x} - 5 = 0$

Remove unnecessary parentheses.

$\frac{6 (x + 1) (x - 1)}{x} - 5 = 0$

Step 6

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

$\frac{6 (x + 1) (x - 1)}{x} - 5 \cdot \frac{x}{x} = 0$

Step 7

Combine $- 5$ and $\frac{x}{x}$ .

$\frac{6 (x + 1) (x - 1)}{x} + \frac{- 5 x}{x} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{6 (x + 1) (x - 1) - 5 x}{x} = 0$

Step 9

Simplify the numerator.

Apply the distributive property.

$\frac{(6 x + 6 \cdot 1) (x - 1) - 5 x}{x} = 0$

Multiply $6$ by $1$ .

$\frac{(6 x + 6) (x - 1) - 5 x}{x} = 0$

Expand $(6 x + 6) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{6 x (x - 1) + 6 (x - 1) - 5 x}{x} = 0$

Apply the distributive property.

$\frac{6 x \cdot x + 6 x \cdot - 1 + 6 (x - 1) - 5 x}{x} = 0$

Apply the distributive property.

$\frac{6 x \cdot x + 6 x \cdot - 1 + 6 x + 6 \cdot - 1 - 5 x}{x} = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{6 (x \cdot x) + 6 x \cdot - 1 + 6 x + 6 \cdot - 1 - 5 x}{x} = 0$

Multiply $x$ by $x$ .

$\frac{6 x^{2} + 6 x \cdot - 1 + 6 x + 6 \cdot - 1 - 5 x}{x} = 0$

Multiply $- 1$ by $6$ .

$\frac{6 x^{2} - 6 x + 6 x + 6 \cdot - 1 - 5 x}{x} = 0$

Multiply $6$ by $- 1$ .

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

Add $- 6 x$ and $6 x$ .

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

Add $6 x^{2}$ and $0$ .

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

Reorder terms.

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

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 $a \cdot c = 6 \cdot - 6 = - 36$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

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

Rewrite $- 5$ as $4$ plus $- 9$

$\frac{6 x^{2} + (4 - 9) x - 6}{x} = 0$

Apply the distributive property.

$\frac{6 x^{2} + 4 x - 9 x - 6}{x} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(6 x^{2} + 4 x) - 9 x - 6}{x} = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $3 x + 2$ .

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

Step 10

Set the numerator equal to zero.

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

Step 11

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 $0$ .

$3 x + 2 = 0$

$2 x - 3 = 0$

Set $3 x + 2$ equal to $0$ and solve for $x$ .

Set $3 x + 2$ equal to $0$ .

$3 x + 2 = 0$

Solve $3 x + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$3 x = - 2$

Divide each term in $3 x = - 2$ by $3$ and simplify.

Divide each term in $3 x = - 2$ by $3$ .

$\frac{3 x}{3} = \frac{- 2}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 2}{3}$

Divide $x$ by $1$ .

$x = \frac{- 2}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{2}{3}$

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

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

Solve by Factoring 6x-5=6/x

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