Solve 3x+4/x+6=3x-2/x-3-23x/x^2+3x-18

$$\frac{ 3x+4 }{ x+6 } = \frac{ 3x-2 }{ x-3 } - \frac{ 23x }{ { x }^{ 2 } +3x-18 }$$

Solve for x

$$x=0$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -6,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+6\right), the least common multiple of x+6,x-3,x^{2}+3x-18.

$$\left(x-3\right)\left(3x+4\right)=\left(x+6\right)\left(3x-2\right)-23x$$

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

$$3x^{2}-5x-12=\left(x+6\right)\left(3x-2\right)-23x$$

Use the distributive property to multiply x+6 by 3x-2 and combine like terms.

$$3x^{2}-5x-12=3x^{2}+16x-12-23x$$

Combine 16x and -23x to get -7x.

$$3x^{2}-5x-12=3x^{2}-7x-12$$

Subtract 3x^{2} from both sides.

$$3x^{2}-5x-12-3x^{2}=-7x-12$$

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

$$-5x-12=-7x-12$$

Add 7x to both sides.

$$-5x-12+7x=-12$$

Combine -5x and 7x to get 2x.

$$2x-12=-12$$

Add 12 to both sides.

$$2x=-12+12$$

Add -12 and 12 to get 0.

$$2x=0$$

Product of two numbers is equal to 0 if at least one of them is 0. Since 2 is not equal to 0, x must be equal to 0.

$$x=0$$