Solve x+22/(x+4)(x-2)=A/x+4+B/x-2

$$\frac { x + 22 } { ( x + 4 ) ( x - 2 ) } = \frac { A } { x + 4 } + \frac { B } { x - 2 }$$

Solve for A

$$A=-\frac{22-4B+x-Bx}{2-x}$$

$$x\neq -4\text{ and }x\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x+4\right), the least common multiple of \left(x+4\right)\left(x-2\right),x+4,x-2.

$$x+22=\left(x-2\right)A+\left(x+4\right)B$$

Use the distributive property to multiply x-2 by A.

$$x+22=xA-2A+\left(x+4\right)B$$

Use the distributive property to multiply x+4 by B.

$$x+22=xA-2A+xB+4B$$

Swap sides so that all variable terms are on the left hand side.

$$xA-2A+xB+4B=x+22$$

Subtract xB from both sides.

$$xA-2A+4B=x+22-xB$$

Subtract 4B from both sides.

$$xA-2A=x+22-xB-4B$$

Combine all terms containing A.

$$\left(x-2\right)A=x+22-xB-4B$$

The equation is in standard form.

$$\left(x-2\right)A=22-4B+x-Bx$$

Divide both sides by x-2.

$$\frac{\left(x-2\right)A}{x-2}=\frac{22-4B+x-Bx}{x-2}$$

Dividing by x-2 undoes the multiplication by x-2.

$$A=\frac{22-4B+x-Bx}{x-2}$$

Solve for B

$$B=\frac{22+2A+x-Ax}{x+4}$$

$$x\neq -4\text{ and }x\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(x+4\right), the least common multiple of \left(x+4\right)\left(x-2\right),x+4,x-2.

$$x+22=\left(x-2\right)A+\left(x+4\right)B$$

Use the distributive property to multiply x-2 by A.

$$x+22=xA-2A+\left(x+4\right)B$$

Use the distributive property to multiply x+4 by B.

$$x+22=xA-2A+xB+4B$$

Swap sides so that all variable terms are on the left hand side.

$$xA-2A+xB+4B=x+22$$

Subtract xA from both sides.

$$-2A+xB+4B=x+22-xA$$

Add 2A to both sides.

$$xB+4B=x+22-xA+2A$$

Combine all terms containing B.

$$\left(x+4\right)B=x+22-xA+2A$$

The equation is in standard form.

$$\left(x+4\right)B=22+2A+x-Ax$$

Divide both sides by x+4.

$$\frac{\left(x+4\right)B}{x+4}=\frac{22+2A+x-Ax}{x+4}$$

Dividing by x+4 undoes the multiplication by x+4.

$$B=\frac{22+2A+x-Ax}{x+4}$$