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

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

Solve for x

$$x=-3$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -5,-4,-2,-1 since division by zero is not defined. Multiply both sides of the equation by \left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+5\right), the least common multiple of x+1,x+2,x+4,x+5.

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

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

$$\left(x^{2}+6x+8\right)\left(x+5\right)x-\left(x+1\right)\left(x+4\right)\left(x+5\right)\left(x+1\right)=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+6x+8 by x+5 and combine like terms.

$$\left(x^{3}+11x^{2}+38x+40\right)x-\left(x+1\right)\left(x+4\right)\left(x+5\right)\left(x+1\right)=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Use the distributive property to multiply x^{3}+11x^{2}+38x+40 by x.

$$x^{4}+11x^{3}+38x^{2}+40x-\left(x+1\right)\left(x+4\right)\left(x+5\right)\left(x+1\right)=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

$$x^{4}+11x^{3}+38x^{2}+40x-\left(x+1\right)^{2}\left(x+4\right)\left(x+5\right)=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{4}+11x^{3}+38x^{2}+40x-\left(x^{2}+2x+1\right)\left(x+4\right)\left(x+5\right)=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Use the distributive property to multiply x^{2}+2x+1 by x+4 and combine like terms.

$$x^{4}+11x^{3}+38x^{2}+40x-\left(x^{3}+6x^{2}+9x+4\right)\left(x+5\right)=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Use the distributive property to multiply x^{3}+6x^{2}+9x+4 by x+5 and combine like terms.

$$x^{4}+11x^{3}+38x^{2}+40x-\left(x^{4}+11x^{3}+39x^{2}+49x+20\right)=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

To find the opposite of x^{4}+11x^{3}+39x^{2}+49x+20, find the opposite of each term.

$$x^{4}+11x^{3}+38x^{2}+40x-x^{4}-11x^{3}-39x^{2}-49x-20=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Combine x^{4} and -x^{4} to get 0.

$$11x^{3}+38x^{2}+40x-11x^{3}-39x^{2}-49x-20=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Combine 11x^{3} and -11x^{3} to get 0.

$$38x^{2}+40x-39x^{2}-49x-20=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Combine 38x^{2} and -39x^{2} to get -x^{2}.

$$-x^{2}+40x-49x-20=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Combine 40x and -49x to get -9x.

$$-x^{2}-9x-20=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)\left(x+4\right)$$

Multiply x+4 and x+4 to get \left(x+4\right)^{2}.

$$-x^{2}-9x-20=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)^{2}$$

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

$$-x^{2}-9x-20=\left(x^{2}+3x+2\right)\left(x+5\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)^{2}$$

Use the distributive property to multiply x^{2}+3x+2 by x+5 and combine like terms.

$$-x^{2}-9x-20=\left(x^{3}+8x^{2}+17x+10\right)\left(x+3\right)-\left(x+1\right)\left(x+2\right)\left(x+4\right)^{2}$$

Use the distributive property to multiply x^{3}+8x^{2}+17x+10 by x+3 and combine like terms.

$$-x^{2}-9x-20=x^{4}+11x^{3}+41x^{2}+61x+30-\left(x+1\right)\left(x+2\right)\left(x+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$-x^{2}-9x-20=x^{4}+11x^{3}+41x^{2}+61x+30-\left(x+1\right)\left(x+2\right)\left(x^{2}+8x+16\right)$$

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

$$-x^{2}-9x-20=x^{4}+11x^{3}+41x^{2}+61x+30-\left(x^{2}+3x+2\right)\left(x^{2}+8x+16\right)$$

Use the distributive property to multiply x^{2}+3x+2 by x^{2}+8x+16 and combine like terms.

$$-x^{2}-9x-20=x^{4}+11x^{3}+41x^{2}+61x+30-\left(x^{4}+11x^{3}+42x^{2}+64x+32\right)$$

To find the opposite of x^{4}+11x^{3}+42x^{2}+64x+32, find the opposite of each term.

$$-x^{2}-9x-20=x^{4}+11x^{3}+41x^{2}+61x+30-x^{4}-11x^{3}-42x^{2}-64x-32$$

Combine x^{4} and -x^{4} to get 0.

$$-x^{2}-9x-20=11x^{3}+41x^{2}+61x+30-11x^{3}-42x^{2}-64x-32$$

Combine 11x^{3} and -11x^{3} to get 0.

$$-x^{2}-9x-20=41x^{2}+61x+30-42x^{2}-64x-32$$

Combine 41x^{2} and -42x^{2} to get -x^{2}.

$$-x^{2}-9x-20=-x^{2}+61x+30-64x-32$$

Combine 61x and -64x to get -3x.

$$-x^{2}-9x-20=-x^{2}-3x+30-32$$

Subtract 32 from 30 to get -2.

$$-x^{2}-9x-20=-x^{2}-3x-2$$

Add x^{2} to both sides.

$$-x^{2}-9x-20+x^{2}=-3x-2$$

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

$$-9x-20=-3x-2$$

Add 3x to both sides.

$$-9x-20+3x=-2$$

Combine -9x and 3x to get -6x.

$$-6x-20=-2$$

Add 20 to both sides.

$$-6x=-2+20$$

Add -2 and 20 to get 18.

$$-6x=18$$

Divide both sides by -6.

$$x=\frac{18}{-6}$$

Divide 18 by -6 to get -3.

$$x=-3$$