Solve 1/x+2+d=1/x+3

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

Solve for d

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

$$x\neq -2\text{ and }x\neq -3$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+2\right)\left(x+3\right), the least common multiple of x+2,x+3.

$$x+3+\left(x+2\right)\left(x+3\right)d=x+2$$

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$x+3+\left(x^{2}+5x+6\right)d=x+2$$

Use the distributive property to multiply x^{2}+5x+6 by d.

$$x+3+x^{2}d+5xd+6d=x+2$$

Subtract x from both sides.

$$3+x^{2}d+5xd+6d=x+2-x$$

Combine x and -x to get 0.

$$3+x^{2}d+5xd+6d=2$$

Subtract 3 from both sides.

$$x^{2}d+5xd+6d=2-3$$

Subtract 3 from 2 to get -1.

$$x^{2}d+5xd+6d=-1$$

Combine all terms containing d.

$$\left(x^{2}+5x+6\right)d=-1$$

Divide both sides by x^{2}+5x+6.

$$\frac{\left(x^{2}+5x+6\right)d}{x^{2}+5x+6}=-\frac{1}{x^{2}+5x+6}$$

Dividing by x^{2}+5x+6 undoes the multiplication by x^{2}+5x+6.

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

Divide -1 by x^{2}+5x+6.

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

Solve for x (complex solution)

$$x=\frac{\sqrt{d^{2}-4d}}{2d}-\frac{5}{2}$$

$$x=-\frac{\sqrt{d^{2}-4d}}{2d}-\frac{5}{2}\text{, }d\neq 0$$

Solve for x

$$x=\frac{\sqrt{d^{2}-4d}}{2d}-\frac{5}{2}$$

$$x=-\frac{\sqrt{d^{2}-4d}}{2d}-\frac{5}{2}\text{, }d<0\text{ or }d\geq 4$$