Solve 3x/x+2+2x/x-7=5x+6/x+2

$$\frac { 3 x } { x + 2 } + \frac { 2 x } { x - 7 } = \frac { 5 x + 6 } { x + 2 }$$

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -2,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x+2\right), the least common multiple of x+2,x-7.

$$\left(x-7\right)\times 3x+\left(x+2\right)\times 2x=\left(x-7\right)\left(5x+6\right)$$

Use the distributive property to multiply x-7 by 3.

$$\left(3x-21\right)x+\left(x+2\right)\times 2x=\left(x-7\right)\left(5x+6\right)$$

Use the distributive property to multiply 3x-21 by x.

$$3x^{2}-21x+\left(x+2\right)\times 2x=\left(x-7\right)\left(5x+6\right)$$

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

$$3x^{2}-21x+\left(2x+4\right)x=\left(x-7\right)\left(5x+6\right)$$

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

$$3x^{2}-21x+2x^{2}+4x=\left(x-7\right)\left(5x+6\right)$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$5x^{2}-21x+4x=\left(x-7\right)\left(5x+6\right)$$

Combine -21x and 4x to get -17x.

$$5x^{2}-17x=\left(x-7\right)\left(5x+6\right)$$

Use the distributive property to multiply x-7 by 5x+6 and combine like terms.

$$5x^{2}-17x=5x^{2}-29x-42$$

Subtract 5x^{2} from both sides.

$$5x^{2}-17x-5x^{2}=-29x-42$$

Combine 5x^{2} and -5x^{2} to get 0.

$$-17x=-29x-42$$

Add 29x to both sides.

$$-17x+29x=-42$$

Combine -17x and 29x to get 12x.

$$12x=-42$$

Divide both sides by 12.

$$x=\frac{-42}{12}$$

Reduce the fraction \frac{-42}{12} to lowest terms by extracting and canceling out 6.

$$x=-\frac{7}{2}$$