Solve by Factoring 2/(x-5)-1/(x+1)=1

$\frac{2}{x - 5} - \frac{1}{x + 1} = 1$

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

Simplify $\frac{2}{x - 5} - \frac{1}{x + 1} - 1$ .

To write $\frac{2}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

To write $- \frac{1}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

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

Write each expression with a common denominator of $(x - 5) (x + 1)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{x - 5}$ by $\frac{x + 1}{x + 1}$ .

$\frac{2 (x + 1)}{(x - 5) (x + 1)} - \frac{1}{x + 1} \cdot \frac{x - 5}{x - 5} - 1 = 0$

Multiply $\frac{1}{x + 1}$ by $\frac{x - 5}{x - 5}$ .

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

Reorder the factors of $(x - 5) (x + 1)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

$\frac{2 x + 2 \cdot 1 - (x - 5)}{(x + 1) (x - 5)} - 1 = 0$

Multiply $2$ by $1$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 5$ .

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

Subtract $x$ from $2 x$ .

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

Add $2$ and $5$ .

$\frac{x + 7}{(x + 1) (x - 5)} - 1 = 0$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{(x + 1) (x - 5)}{(x + 1) (x - 5)}$ .

$\frac{x + 7}{(x + 1) (x - 5)} - 1 \cdot \frac{(x + 1) (x - 5)}{(x + 1) (x - 5)} = 0$

Combine $- 1$ and $\frac{(x + 1) (x - 5)}{(x + 1) (x - 5)}$ .

$\frac{x + 7}{(x + 1) (x - 5)} + \frac{- ((x + 1) (x - 5))}{(x + 1) (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{x + 7 - ((x + 1) (x - 5))}{(x + 1) (x - 5)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{x + 7 + (- x - 1 \cdot 1) (x - 5)}{(x + 1) (x - 5)} = 0$

Multiply $- 1$ by $1$ .

$\frac{x + 7 + (- x - 1) (x - 5)}{(x + 1) (x - 5)} = 0$

Expand $(- x - 1) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{x + 7 - x (x - 5) - 1 (x - 5)}{(x + 1) (x - 5)} = 0$

Apply the distributive property.

$\frac{x + 7 - x \cdot x - x \cdot - 5 - 1 (x - 5)}{(x + 1) (x - 5)} = 0$

Apply the distributive property.

$\frac{x + 7 - x \cdot x - x \cdot - 5 - 1 x - 1 \cdot - 5}{(x + 1) (x - 5)} = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{x + 7 - (x \cdot x) - x \cdot - 5 - 1 x - 1 \cdot - 5}{(x + 1) (x - 5)} = 0$

Multiply $x$ by $x$ .

$\frac{x + 7 - x^{2} - x \cdot - 5 - 1 x - 1 \cdot - 5}{(x + 1) (x - 5)} = 0$

Multiply $- 5$ by $- 1$ .

$\frac{x + 7 - x^{2} + 5 x - 1 x - 1 \cdot - 5}{(x + 1) (x - 5)} = 0$

Rewrite $- 1 x$ as $- x$ .

$\frac{x + 7 - x^{2} + 5 x - x - 1 \cdot - 5}{(x + 1) (x - 5)} = 0$

Multiply $- 1$ by $- 5$ .

$\frac{x + 7 - x^{2} + 5 x - x + 5}{(x + 1) (x - 5)} = 0$

Subtract $x$ from $5 x$ .

$\frac{x + 7 - x^{2} + 4 x + 5}{(x + 1) (x - 5)} = 0$

Add $x$ and $4 x$ .

$\frac{5 x + 7 - x^{2} + 5}{(x + 1) (x - 5)} = 0$

Add $7$ and $5$ .

$\frac{5 x - x^{2} + 12}{(x + 1) (x - 5)} = 0$

Reorder terms.

$\frac{- x^{2} + 5 x + 12}{(x + 1) (x - 5)} = 0$

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

$\frac{- (x^{2}) + 5 x + 12}{(x + 1) (x - 5)} = 0$

Factor $- 1$ out of $5 x$ .

$\frac{- (x^{2}) - (- 5 x) + 12}{(x + 1) (x - 5)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 5 x)$ .

$\frac{- (x^{2} - 5 x) + 12}{(x + 1) (x - 5)} = 0$

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

$\frac{- (x^{2} - 5 x) - 1 (- 12)}{(x + 1) (x - 5)} = 0$

Factor $- 1$ out of $- (x^{2} - 5 x) - 1 (- 12)$ .

$\frac{- (x^{2} - 5 x - 12)}{(x + 1) (x - 5)} = 0$

Rewrite $- (x^{2} - 5 x - 12)$ as $- 1 (x^{2} - 5 x - 12)$ .

$\frac{- 1 (x^{2} - 5 x - 12)}{(x + 1) (x - 5)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 5 x - 12}{(x + 1) (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 5 x - 12 = 0$

Step 4

Solve the equation for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 5$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 48}}{2 \cdot 1}$

Add $25$ and $48$ .

$x = \frac{5 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{73}}{2}$

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

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 48}}{2 \cdot 1}$

Add $25$ and $48$ .

$x = \frac{5 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{73}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{73}}{2}$

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

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 48}}{2 \cdot 1}$

Add $25$ and $48$ .

$x = \frac{5 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{73}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{73}}{2}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{73}}{2}, \frac{5 - \sqrt{73}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{73}}{2}, \frac{5 - \sqrt{73}}{2}$

Decimal Form:

$x = 6.77200187 \dots, - 1.77200187 \dots$

Solve by Factoring 2/(x-5)-1/(x+1)=1

Related Questions

Support LCMGCF.com by disabling your adblocker.