Solve 6/2x+3+4/x-2=7/x+6

$$\frac { 6 } { 2 x + 3 } + \frac { 4 } { x - 2 } = \frac { 7 } { x + 6 }$$

Solve for x

$$x=-\frac{6}{13}\approx -0.461538462$$

Steps for Solving Linear Equation

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

$$\left(-2+x\right)\left(x+6\right)\times 6+\left(2x+3\right)\left(x+6\right)\times 4=\left(-2+x\right)\left(2x+3\right)\times 7$$

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

$$\left(4x-12+x^{2}\right)\times 6+\left(2x+3\right)\left(x+6\right)\times 4=\left(-2+x\right)\left(2x+3\right)\times 7$$

Use the distributive property to multiply 4x-12+x^{2} by 6.

$$24x-72+6x^{2}+\left(2x+3\right)\left(x+6\right)\times 4=\left(-2+x\right)\left(2x+3\right)\times 7$$

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

$$24x-72+6x^{2}+\left(2x^{2}+15x+18\right)\times 4=\left(-2+x\right)\left(2x+3\right)\times 7$$

Use the distributive property to multiply 2x^{2}+15x+18 by 4.

$$24x-72+6x^{2}+8x^{2}+60x+72=\left(-2+x\right)\left(2x+3\right)\times 7$$

Combine 6x^{2} and 8x^{2} to get 14x^{2}.

$$24x-72+14x^{2}+60x+72=\left(-2+x\right)\left(2x+3\right)\times 7$$

Combine 24x and 60x to get 84x.

$$84x-72+14x^{2}+72=\left(-2+x\right)\left(2x+3\right)\times 7$$

Add -72 and 72 to get 0.

$$84x+14x^{2}=\left(-2+x\right)\left(2x+3\right)\times 7$$

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

$$84x+14x^{2}=\left(-x-6+2x^{2}\right)\times 7$$

Use the distributive property to multiply -x-6+2x^{2} by 7.

$$84x+14x^{2}=-7x-42+14x^{2}$$

Add 7x to both sides.

$$84x+14x^{2}+7x=-42+14x^{2}$$

Combine 84x and 7x to get 91x.

$$91x+14x^{2}=-42+14x^{2}$$

Subtract 14x^{2} from both sides.

$$91x+14x^{2}-14x^{2}=-42$$

Combine 14x^{2} and -14x^{2} to get 0.

$$91x=-42$$

Divide both sides by 91.

$$x=\frac{-42}{91}$$

Reduce the fraction \frac{-42}{91} to lowest terms by extracting and canceling out 7.

$$x=-\frac{6}{13}$$