Solve x+4/2x^2-5x+2+x+1/2x^2-7x+3=2x+5/2x^2-7x+3

$$\frac { x + 4 } { 2 x ^ { 2 } - 5 x + 2 } + \frac { x + 1 } { 2 x ^ { 2 } - 7 x + 3 } = \frac { 2 x + 5 } { 2 x ^ { 2 } - 7 x + 3 }$$

Solve for x

$$x=-4$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values \frac{1}{2},2,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(\frac{1}{2}x-1\right)\left(2x-1\right), the least common multiple of 2x^{2}-5x+2,2x^{2}-7x+3.

$$\left(x-3\right)\left(x+4\right)+\left(-2+x\right)\left(x+1\right)=\left(-2+x\right)\left(2x+5\right)$$

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

$$x^{2}+x-12+\left(-2+x\right)\left(x+1\right)=\left(-2+x\right)\left(2x+5\right)$$

Use the distributive property to multiply -2+x by x+1 and combine like terms.

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

Combine x and -x to get 0.

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

Subtract 2 from -12 to get -14.

$$x^{2}-14+x^{2}=\left(-2+x\right)\left(2x+5\right)$$

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

$$2x^{2}-14=\left(-2+x\right)\left(2x+5\right)$$

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

$$2x^{2}-14=x-10+2x^{2}$$

Subtract x from both sides.

$$2x^{2}-14-x=-10+2x^{2}$$

Subtract 2x^{2} from both sides.

$$2x^{2}-14-x-2x^{2}=-10$$

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

$$-14-x=-10$$

Add 14 to both sides.

$$-x=-10+14$$

Add -10 and 14 to get 4.

$$-x=4$$

Multiply both sides by -1.

$$x=-4$$