Solve 5/x+1-1=10-x/x-1

$$\frac { 5 } { x + 1 } - 1 = \frac { 10 - x } { x - 1 }$$

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+1\right), the least common multiple of x+1,x-1.

$$\left(x-1\right)\times 5+\left(x-1\right)\left(x+1\right)\left(-1\right)=\left(x+1\right)\left(10-x\right)$$

Use the distributive property to multiply x-1 by 5.

$$5x-5+\left(x-1\right)\left(x+1\right)\left(-1\right)=\left(x+1\right)\left(10-x\right)$$

Use the distributive property to multiply x-1 by x+1 and combine like terms.

$$5x-5+\left(x^{2}-1\right)\left(-1\right)=\left(x+1\right)\left(10-x\right)$$

Use the distributive property to multiply x^{2}-1 by -1.

$$5x-5-x^{2}+1=\left(x+1\right)\left(10-x\right)$$

Add -5 and 1 to get -4.

$$5x-4-x^{2}=\left(x+1\right)\left(10-x\right)$$

Use the distributive property to multiply x+1 by 10-x and combine like terms.

$$5x-4-x^{2}=9x-x^{2}+10$$

Subtract 9x from both sides.

$$5x-4-x^{2}-9x=-x^{2}+10$$

Combine 5x and -9x to get -4x.

$$-4x-4-x^{2}=-x^{2}+10$$

Add x^{2} to both sides.

$$-4x-4-x^{2}+x^{2}=10$$

Combine -x^{2} and x^{2} to get 0.

$$-4x-4=10$$

Add 4 to both sides.

$$-4x=10+4$$

Add 10 and 4 to get 14.

$$-4x=14$$

Divide both sides by -4.

$$x=\frac{14}{-4}$$

Reduce the fraction \frac{14}{-4} to lowest terms by extracting and canceling out 2.

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