Solve 5(x-2)/x+2-2(x-3)/x+3=3

$$\frac { 5 ( x - 2 ) } { x + 2 } - \frac { 2 ( x - 3 ) } { x + 3 } = 3$$

Solve for x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply x+3 by 5.

$$\left(5x+15\right)\left(x-2\right)-\left(x+2\right)\times 2\left(x-3\right)=3\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 5x+15 by x-2 and combine like terms.

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

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

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

Use the distributive property to multiply 2x+4 by x-3 and combine like terms.

$$5x^{2}+5x-30-\left(2x^{2}-2x-12\right)=3\left(x+2\right)\left(x+3\right)$$

To find the opposite of 2x^{2}-2x-12, find the opposite of each term.

$$5x^{2}+5x-30-2x^{2}+2x+12=3\left(x+2\right)\left(x+3\right)$$

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

$$3x^{2}+5x-30+2x+12=3\left(x+2\right)\left(x+3\right)$$

Combine 5x and 2x to get 7x.

$$3x^{2}+7x-30+12=3\left(x+2\right)\left(x+3\right)$$

Add -30 and 12 to get -18.

$$3x^{2}+7x-18=3\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply 3 by x+2.

$$3x^{2}+7x-18=\left(3x+6\right)\left(x+3\right)$$

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

$$3x^{2}+7x-18=3x^{2}+15x+18$$

Subtract 3x^{2} from both sides.

$$3x^{2}+7x-18-3x^{2}=15x+18$$

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

$$7x-18=15x+18$$

Subtract 15x from both sides.

$$7x-18-15x=18$$

Combine 7x and -15x to get -8x.

$$-8x-18=18$$

Add 18 to both sides.

$$-8x=18+18$$

Add 18 and 18 to get 36.

$$-8x=36$$

Divide both sides by -8.

$$x=\frac{36}{-8}$$

Reduce the fraction \frac{36}{-8} to lowest terms by extracting and canceling out 4.

$$x=-\frac{9}{2}$$