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

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

Solve for x

$$x = -\frac{11}{8} = -1\frac{3}{8} = -1.375$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -\frac{2}{3},5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(3x+2\right), the least common multiple of x-5,3x+2.

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

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

$$3x^{2}+20x+12-\left(x-5\right)=\left(3x+2\right)\left(x+3\right)$$

To find the opposite of x-5, find the opposite of each term.

$$3x^{2}+20x+12-x+5=\left(3x+2\right)\left(x+3\right)$$

Combine 20x and -x to get 19x.

$$3x^{2}+19x+12+5=\left(3x+2\right)\left(x+3\right)$$

Add 12 and 5 to get 17.

$$3x^{2}+19x+17=\left(3x+2\right)\left(x+3\right)$$

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

$$3x^{2}+19x+17=3x^{2}+11x+6$$

Subtract 3x^{2} from both sides.

$$3x^{2}+19x+17-3x^{2}=11x+6$$

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

$$19x+17=11x+6$$

Subtract 11x from both sides.

$$19x+17-11x=6$$

Combine 19x and -11x to get 8x.

$$8x+17=6$$

Subtract 17 from both sides.

$$8x=6-17$$

Subtract 17 from 6 to get -11.

$$8x=-11$$

Divide both sides by 8.

$$x=\frac{-11}{8}$$

Fraction \frac{-11}{8} can be rewritten as -\frac{11}{8} by extracting the negative sign.

$$x=-\frac{11}{8}$$