Solve x+1/2/frac{x-1{2}}-x-1/2/frac{x+1{2}}=2frac{2}{3}

$$\frac { \frac { x + 1 } { 2 } } { \frac { x - 1 } { 2 } } - \frac { \frac { x - 1 } { 2 } } { \frac { x + 1 } { 2 } } = 2 \frac { 2 } { 3 }$$

Solve for x

$$x=2$$

$$x=-\frac{1}{2}=-0.5$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$6\left(x-1\right)^{-1}\times \frac{x+1}{2}-6\left(x+1\right)^{-1}\times \frac{x-1}{2}=2\times 3+2$$

Cancel out 2, the greatest common factor in 6 and 2.

$$3\left(x+1\right)\left(x-1\right)^{-1}-6\left(x+1\right)^{-1}\times \frac{x-1}{2}=2\times 3+2$$

Cancel out 2, the greatest common factor in 6 and 2.

$$3\left(x+1\right)\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=2\times 3+2$$

Multiply 2 and 3 to get 6.

$$3\left(x+1\right)\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=6+2$$

Add 6 and 2 to get 8.

$$3\left(x+1\right)\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=8$$

Use the distributive property to multiply 3 by x+1.

$$\left(3x+3\right)\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=8$$

Use the distributive property to multiply 3x+3 by \left(x-1\right)^{-1}.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=8$$

Use the distributive property to multiply 3 by x-1.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-\left(3x-3\right)\left(x+1\right)^{-1}=8$$

Use the distributive property to multiply 3x-3 by \left(x+1\right)^{-1}.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-\left(3x\left(x+1\right)^{-1}-3\left(x+1\right)^{-1}\right)=8$$

To find the opposite of 3x\left(x+1\right)^{-1}-3\left(x+1\right)^{-1}, find the opposite of each term.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-3x\left(x+1\right)^{-1}+3\left(x+1\right)^{-1}=8$$

Subtract 8 from both sides.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-3x\left(x+1\right)^{-1}+3\left(x+1\right)^{-1}-8=0$$

Reorder the terms.

$$3\times \frac{1}{x-1}x-3\times \frac{1}{x+1}x-8+3\times \frac{1}{x+1}+3\times \frac{1}{x-1}=0$$

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.

$$3\left(x+1\right)\times 1x-3\left(x-1\right)x+\left(x-1\right)\left(x+1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Multiply 3 and 1 to get 3.

$$3\left(x+1\right)x-3\left(x-1\right)x+\left(x-1\right)\left(x+1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Use the distributive property to multiply 3 by x+1.

$$\left(3x+3\right)x-3\left(x-1\right)x+\left(x-1\right)\left(x+1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Use the distributive property to multiply 3x+3 by x.

$$3x^{2}+3x-3\left(x-1\right)x+\left(x-1\right)\left(x+1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Use the distributive property to multiply -3 by x-1.

$$3x^{2}+3x+\left(-3x+3\right)x+\left(x-1\right)\left(x+1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Use the distributive property to multiply -3x+3 by x.

$$3x^{2}+3x-3x^{2}+3x+\left(x-1\right)\left(x+1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Combine 3x^{2} and -3x^{2} to get 0.

$$3x+3x+\left(x-1\right)\left(x+1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Combine 3x and 3x to get 6x.

$$6x+\left(x-1\right)\left(x+1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

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

$$6x+\left(x^{2}-1\right)\left(-8\right)+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Use the distributive property to multiply x^{2}-1 by -8.

$$6x-8x^{2}+8+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=0$$

Multiply 3 and 1 to get 3.

$$6x-8x^{2}+8+3\left(x-1\right)+3\left(x+1\right)\times 1=0$$

Use the distributive property to multiply 3 by x-1.

$$6x-8x^{2}+8+3x-3+3\left(x+1\right)\times 1=0$$

Combine 6x and 3x to get 9x.

$$9x-8x^{2}+8-3+3\left(x+1\right)\times 1=0$$

Subtract 3 from 8 to get 5.

$$9x-8x^{2}+5+3\left(x+1\right)\times 1=0$$

Multiply 3 and 1 to get 3.

$$9x-8x^{2}+5+3\left(x+1\right)=0$$

Use the distributive property to multiply 3 by x+1.

$$9x-8x^{2}+5+3x+3=0$$

Combine 9x and 3x to get 12x.

$$12x-8x^{2}+5+3=0$$

Add 5 and 3 to get 8.

$$12x-8x^{2}+8=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-8x^{2}+12x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 12 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-8\right)\times 8}}{2\left(-8\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-8\right)\times 8}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-12±\sqrt{144+32\times 8}}{2\left(-8\right)}$$

Multiply 32 times 8.

$$x=\frac{-12±\sqrt{144+256}}{2\left(-8\right)}$$

Add 144 to 256.

$$x=\frac{-12±\sqrt{400}}{2\left(-8\right)}$$

Take the square root of 400.

$$x=\frac{-12±20}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-12±20}{-16}$$

Now solve the equation x=\frac{-12±20}{-16} when ± is plus. Add -12 to 20.

$$x=\frac{8}{-16}$$

Reduce the fraction \frac{8}{-16} to lowest terms by extracting and canceling out 8.

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

Now solve the equation x=\frac{-12±20}{-16} when ± is minus. Subtract 20 from -12.

$$x=-\frac{32}{-16}$$

Divide -32 by -16.

$$x=2$$

The equation is now solved.

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

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$6\left(x-1\right)^{-1}\times \frac{x+1}{2}-6\left(x+1\right)^{-1}\times \frac{x-1}{2}=2\times 3+2$$

Cancel out 2, the greatest common factor in 6 and 2.

$$3\left(x+1\right)\left(x-1\right)^{-1}-6\left(x+1\right)^{-1}\times \frac{x-1}{2}=2\times 3+2$$

Cancel out 2, the greatest common factor in 6 and 2.

$$3\left(x+1\right)\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=2\times 3+2$$

Multiply 2 and 3 to get 6.

$$3\left(x+1\right)\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=6+2$$

Add 6 and 2 to get 8.

$$3\left(x+1\right)\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=8$$

Use the distributive property to multiply 3 by x+1.

$$\left(3x+3\right)\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=8$$

Use the distributive property to multiply 3x+3 by \left(x-1\right)^{-1}.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-3\left(x-1\right)\left(x+1\right)^{-1}=8$$

Use the distributive property to multiply 3 by x-1.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-\left(3x-3\right)\left(x+1\right)^{-1}=8$$

Use the distributive property to multiply 3x-3 by \left(x+1\right)^{-1}.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-\left(3x\left(x+1\right)^{-1}-3\left(x+1\right)^{-1}\right)=8$$

To find the opposite of 3x\left(x+1\right)^{-1}-3\left(x+1\right)^{-1}, find the opposite of each term.

$$3x\left(x-1\right)^{-1}+3\left(x-1\right)^{-1}-3x\left(x+1\right)^{-1}+3\left(x+1\right)^{-1}=8$$

Reorder the terms.

$$3\times \frac{1}{x-1}x-3\times \frac{1}{x+1}x+3\times \frac{1}{x+1}+3\times \frac{1}{x-1}=8$$

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.

$$3\left(x+1\right)\times 1x-3\left(x-1\right)x+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Multiply 3 and 1 to get 3.

$$3\left(x+1\right)x-3\left(x-1\right)x+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 3 by x+1.

$$\left(3x+3\right)x-3\left(x-1\right)x+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 3x+3 by x.

$$3x^{2}+3x-3\left(x-1\right)x+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply -3 by x-1.

$$3x^{2}+3x+\left(-3x+3\right)x+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply -3x+3 by x.

$$3x^{2}+3x-3x^{2}+3x+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Combine 3x^{2} and -3x^{2} to get 0.

$$3x+3x+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Combine 3x and 3x to get 6x.

$$6x+3\left(x-1\right)\times 1+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Multiply 3 and 1 to get 3.

$$6x+3\left(x-1\right)+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 3 by x-1.

$$6x+3x-3+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Combine 6x and 3x to get 9x.

$$9x-3+3\left(x+1\right)\times 1=8\left(x-1\right)\left(x+1\right)$$

Multiply 3 and 1 to get 3.

$$9x-3+3\left(x+1\right)=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 3 by x+1.

$$9x-3+3x+3=8\left(x-1\right)\left(x+1\right)$$

Combine 9x and 3x to get 12x.

$$12x-3+3=8\left(x-1\right)\left(x+1\right)$$

Add -3 and 3 to get 0.

$$12x=8\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 8 by x-1.

$$12x=\left(8x-8\right)\left(x+1\right)$$

Use the distributive property to multiply 8x-8 by x+1 and combine like terms.

$$12x=8x^{2}-8$$

Subtract 8x^{2} from both sides.

$$12x-8x^{2}=-8$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-8x^{2}+12x=-8$$

Divide both sides by -8.

$$\frac{-8x^{2}+12x}{-8}=-\frac{8}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{12}{-8}x=-\frac{8}{-8}$$

Reduce the fraction \frac{12}{-8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{3}{2}x=-\frac{8}{-8}$$

Divide -8 by -8.

$$x^{2}-\frac{3}{2}x=1$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=1+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=1+\frac{9}{16}$$

Add 1 to \frac{9}{16}.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{25}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{25}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Simplify.

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

Add \frac{3}{4} to both sides of the equation.

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