Solve 4/x+1-3/x+2=ax+b/(x+1)(x+2)

$$\frac { 4 } { x + 1 } - \frac { 3 } { x + 2 } = \frac { a x + b } { ( x + 1 ) ( x + 2 ) }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{-x+b-5}{x}\text{, }&x\neq 0\text{ and }x\neq -1\text{ and }x\neq -2\\a\in \mathrm{C}\text{, }&x=0\text{ and }b=5\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,\left(x+1\right)\left(x+2\right).

$$\left(x+2\right)\times 4-\left(x+1\right)\times 3=ax+b$$

Use the distributive property to multiply x+2 by 4.

$$4x+8-\left(x+1\right)\times 3=ax+b$$

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

$$4x+8-\left(3x+3\right)=ax+b$$

To find the opposite of 3x+3, find the opposite of each term.

$$4x+8-3x-3=ax+b$$

Combine 4x and -3x to get x.

$$x+8-3=ax+b$$

Subtract 3 from 8 to get 5.

$$x+5=ax+b$$

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

$$ax+b=x+5$$

Subtract b from both sides.

$$ax=x+5-b$$

The equation is in standard form.

$$xa=x-b+5$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{x-b+5}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{x-b+5}{x}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{-x+b-5}{x}\text{, }&x\neq 0\text{ and }x\neq -1\text{ and }x\neq -2\\a\in \mathrm{R}\text{, }&x=0\text{ and }b=5\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,\left(x+1\right)\left(x+2\right).

$$\left(x+2\right)\times 4-\left(x+1\right)\times 3=ax+b$$

Use the distributive property to multiply x+2 by 4.

$$4x+8-\left(x+1\right)\times 3=ax+b$$

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

$$4x+8-\left(3x+3\right)=ax+b$$

To find the opposite of 3x+3, find the opposite of each term.

$$4x+8-3x-3=ax+b$$

Combine 4x and -3x to get x.

$$x+8-3=ax+b$$

Subtract 3 from 8 to get 5.

$$x+5=ax+b$$

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

$$ax+b=x+5$$

Subtract b from both sides.

$$ax=x+5-b$$

The equation is in standard form.

$$xa=x-b+5$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{x-b+5}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{x-b+5}{x}$$

Solve for b

$$b=5+x-ax$$

$$x\neq -1\text{ and }x\neq -2$$

Solution Steps

Multiply both sides of the equation by \left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,\left(x+1\right)\left(x+2\right).

$$\left(x+2\right)\times 4-\left(x+1\right)\times 3=ax+b$$

Use the distributive property to multiply x+2 by 4.

$$4x+8-\left(x+1\right)\times 3=ax+b$$

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

$$4x+8-\left(3x+3\right)=ax+b$$

To find the opposite of 3x+3, find the opposite of each term.

$$4x+8-3x-3=ax+b$$

Combine 4x and -3x to get x.

$$x+8-3=ax+b$$

Subtract 3 from 8 to get 5.

$$x+5=ax+b$$

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

$$ax+b=x+5$$

Subtract ax from both sides.

$$b=x+5-ax$$