Solve x+5/x+1-2x+5/x+2=x-5/3x+3-4x-3/3x+6

$$\frac { x + 5 } { x + 1 } - \frac { 2 x + 5 } { x + 2 } = \frac { x - 5 } { 3 x + 3 } - \frac { 4 x - 3 } { 3 x + 6 }$$

Solve for x

$$x = -\frac{11}{2} = -5\frac{1}{2} = -5.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -2,-1 since division by zero is not defined. Multiply both sides of the equation by 3\left(x+1\right)\left(x+2\right), the least common multiple of x+1,x+2,3x+3,3x+6.

$$\left(3x+6\right)\left(x+5\right)-\left(3x+3\right)\left(2x+5\right)=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Use the distributive property to multiply 3x+6 by x+5 and combine like terms.

$$3x^{2}+21x+30-\left(3x+3\right)\left(2x+5\right)=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Use the distributive property to multiply 3x+3 by 2x+5 and combine like terms.

$$3x^{2}+21x+30-\left(6x^{2}+21x+15\right)=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

To find the opposite of 6x^{2}+21x+15, find the opposite of each term.

$$3x^{2}+21x+30-6x^{2}-21x-15=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Combine 3x^{2} and -6x^{2} to get -3x^{2}.

$$-3x^{2}+21x+30-21x-15=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Combine 21x and -21x to get 0.

$$-3x^{2}+30-15=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Subtract 15 from 30 to get 15.

$$-3x^{2}+15=\left(x+2\right)\left(x-5\right)-\left(x+1\right)\left(4x-3\right)$$

Use the distributive property to multiply x+2 by x-5 and combine like terms.

$$-3x^{2}+15=x^{2}-3x-10-\left(x+1\right)\left(4x-3\right)$$

Use the distributive property to multiply x+1 by 4x-3 and combine like terms.

$$-3x^{2}+15=x^{2}-3x-10-\left(4x^{2}+x-3\right)$$

To find the opposite of 4x^{2}+x-3, find the opposite of each term.

$$-3x^{2}+15=x^{2}-3x-10-4x^{2}-x+3$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}+15=-3x^{2}-3x-10-x+3$$

Combine -3x and -x to get -4x.

$$-3x^{2}+15=-3x^{2}-4x-10+3$$

Add -10 and 3 to get -7.

$$-3x^{2}+15=-3x^{2}-4x-7$$

Add 3x^{2} to both sides.

$$-3x^{2}+15+3x^{2}=-4x-7$$

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

$$15=-4x-7$$

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

$$-4x-7=15$$

Add 7 to both sides.

$$-4x=15+7$$

Add 15 and 7 to get 22.

$$-4x=22$$

Divide both sides by -4.

$$x=\frac{22}{-4}$$

Reduce the fraction \frac{22}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{11}{2}$$