Solve for x -7/x+1=-6x

$- \frac{7}{x} + 1 = - 6 x$

Step 1

Subtract $1$ from both sides of the equation.

$- \frac{7}{x} = - 6 x - 1$

Step 2

Find the LCD of the terms in the equation.

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

$x, 1, 1$

The LCM of one and any expression is the expression.

$x$

Step 3

Multiply each term in $- \frac{7}{x} = - 6 x - 1$ by $x$ to eliminate the fractions.

Multiply each term in $- \frac{7}{x} = - 6 x - 1$ by $x$ .

$- \frac{7}{x} x = - 6 x \cdot x - x$

Simplify the left side.

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{7}{x}$ into the numerator.

$\frac{- 7}{x} x = - 6 x \cdot x - x$

Cancel the common factor.

$\frac{- 7}{x} x = - 6 x \cdot x - x$

Rewrite the expression.

$- 7 = - 6 x \cdot x - x$

Simplify the right side.

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

Move $x$ .

$- 7 = - 6 (x \cdot x) - x$

Multiply $x$ by $x$ .

$- 7 = - 6 x^{2} - x$

Step 4

Solve the equation.

Rewrite the equation as $- 6 x^{2} - x = - 7$ .

$- 6 x^{2} - x = - 7$

Add $7$ to both sides of the equation.

$- 6 x^{2} - x + 7 = 0$

Factor the left side of the equation.

Factor $- 1$ out of $- 6 x^{2} - x + 7$ .

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

$- (6 x^{2}) - x + 7 = 0$

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

$- (6 x^{2}) - (x) + 7 = 0$

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

$- (6 x^{2}) - (x) - 1 \cdot - 7 = 0$

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

$- (6 x^{2} + x) - 1 \cdot - 7 = 0$

Factor $- 1$ out of $- (6 x^{2} + x) - 1 (- 7)$ .

$- (6 x^{2} + x - 7) = 0$

Factor.

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 $a \cdot c = 6 \cdot - 7 = - 42$ and whose sum is $b = 1$ .

Multiply by $1$ .

$- (6 x^{2} + 1 x - 7) = 0$

Rewrite $1$ as $- 6$ plus $7$

$- (6 x^{2} + (- 6 + 7) x - 7) = 0$

Apply the distributive property.

$- (6 x^{2} - 6 x + 7 x - 7) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- ((6 x^{2} - 6 x) + 7 x - 7) = 0$

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

$- (6 x (x - 1) + 7 (x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$- ((x - 1) (6 x + 7)) = 0$

Remove unnecessary parentheses.

$- (x - 1) (6 x + 7) = 0$

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

$x - 1 = 0$

$6 x + 7 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $6 x + 7$ equal to $0$ and solve for $x$ .

Set $6 x + 7$ equal to $0$ .

$6 x + 7 = 0$

Solve $6 x + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

$6 x = - 7$

Divide each term in $6 x = - 7$ by $6$ and simplify.

Divide each term in $6 x = - 7$ by $6$ .

$\frac{6 x}{6} = \frac{- 7}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 7}{6}$

Divide $x$ by $1$ .

$x = \frac{- 7}{6}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{7}{6}$

The final solution is all the values that make $- (x - 1) (6 x + 7) = 0$ true.

$x = 1, - \frac{7}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 1, - \frac{7}{6}$

Decimal Form:

$x = 1, - 1.1\overline{6}$

Mixed Number Form:

$x = 1, - 1 \frac{1}{6}$

Solve for x -7/x+1=-6x

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