Solve x+1/x+2+x+6/x+7=x+2/x+3+x+5/x+6

$$\frac { x + 1 } { x + 2 } + \frac { x + 6 } { x + 7 } = \frac { x + 2 } { x + 3 } + \frac { x + 5 } { x + 6 }$$

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 -7,-6,-3,-2 since division by zero is not defined. Multiply both sides of the equation by \left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+7\right), the least common multiple of x+2,x+7,x+3,x+6.

$$\left(x+3\right)\left(x+6\right)\left(x+7\right)\left(x+1\right)+\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+6\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

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

$$\left(x^{2}+9x+18\right)\left(x+7\right)\left(x+1\right)+\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+6\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Use the distributive property to multiply x^{2}+9x+18 by x+7 and combine like terms.

$$\left(x^{3}+16x^{2}+81x+126\right)\left(x+1\right)+\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+6\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Use the distributive property to multiply x^{3}+16x^{2}+81x+126 by x+1 and combine like terms.

$$x^{4}+17x^{3}+97x^{2}+207x+126+\left(x+2\right)\left(x+3\right)\left(x+6\right)\left(x+6\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Multiply x+6 and x+6 to get \left(x+6\right)^{2}.

$$x^{4}+17x^{3}+97x^{2}+207x+126+\left(x+2\right)\left(x+3\right)\left(x+6\right)^{2}=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{4}+17x^{3}+97x^{2}+207x+126+\left(x+2\right)\left(x+3\right)\left(x^{2}+12x+36\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

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

$$x^{4}+17x^{3}+97x^{2}+207x+126+\left(x^{2}+5x+6\right)\left(x^{2}+12x+36\right)=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Use the distributive property to multiply x^{2}+5x+6 by x^{2}+12x+36 and combine like terms.

$$x^{4}+17x^{3}+97x^{2}+207x+126+x^{4}+17x^{3}+102x^{2}+252x+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}+17x^{3}+97x^{2}+207x+126+17x^{3}+102x^{2}+252x+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Combine 17x^{3} and 17x^{3} to get 34x^{3}.

$$2x^{4}+34x^{3}+97x^{2}+207x+126+102x^{2}+252x+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Combine 97x^{2} and 102x^{2} to get 199x^{2}.

$$2x^{4}+34x^{3}+199x^{2}+207x+126+252x+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Combine 207x and 252x to get 459x.

$$2x^{4}+34x^{3}+199x^{2}+459x+126+216=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Add 126 and 216 to get 342.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=\left(x+2\right)\left(x+6\right)\left(x+7\right)\left(x+2\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

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

$$2x^{4}+34x^{3}+199x^{2}+459x+342=\left(x+2\right)^{2}\left(x+6\right)\left(x+7\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

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

$$2x^{4}+34x^{3}+199x^{2}+459x+342=\left(x^{2}+4x+4\right)\left(x+6\right)\left(x+7\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

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

$$2x^{4}+34x^{3}+199x^{2}+459x+342=\left(x^{3}+10x^{2}+28x+24\right)\left(x+7\right)+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

Use the distributive property to multiply x^{3}+10x^{2}+28x+24 by x+7 and combine like terms.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=x^{4}+17x^{3}+98x^{2}+220x+168+\left(x+2\right)\left(x+3\right)\left(x+7\right)\left(x+5\right)$$

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

$$2x^{4}+34x^{3}+199x^{2}+459x+342=x^{4}+17x^{3}+98x^{2}+220x+168+\left(x^{2}+5x+6\right)\left(x+7\right)\left(x+5\right)$$

Use the distributive property to multiply x^{2}+5x+6 by x+7 and combine like terms.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=x^{4}+17x^{3}+98x^{2}+220x+168+\left(x^{3}+12x^{2}+41x+42\right)\left(x+5\right)$$

Use the distributive property to multiply x^{3}+12x^{2}+41x+42 by x+5 and combine like terms.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=x^{4}+17x^{3}+98x^{2}+220x+168+x^{4}+17x^{3}+101x^{2}+247x+210$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+17x^{3}+98x^{2}+220x+168+17x^{3}+101x^{2}+247x+210$$

Combine 17x^{3} and 17x^{3} to get 34x^{3}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+34x^{3}+98x^{2}+220x+168+101x^{2}+247x+210$$

Combine 98x^{2} and 101x^{2} to get 199x^{2}.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+34x^{3}+199x^{2}+220x+168+247x+210$$

Combine 220x and 247x to get 467x.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+34x^{3}+199x^{2}+467x+168+210$$

Add 168 and 210 to get 378.

$$2x^{4}+34x^{3}+199x^{2}+459x+342=2x^{4}+34x^{3}+199x^{2}+467x+378$$

Subtract 2x^{4} from both sides.

$$2x^{4}+34x^{3}+199x^{2}+459x+342-2x^{4}=34x^{3}+199x^{2}+467x+378$$

Combine 2x^{4} and -2x^{4} to get 0.

$$34x^{3}+199x^{2}+459x+342=34x^{3}+199x^{2}+467x+378$$

Subtract 34x^{3} from both sides.

$$34x^{3}+199x^{2}+459x+342-34x^{3}=199x^{2}+467x+378$$

Combine 34x^{3} and -34x^{3} to get 0.

$$199x^{2}+459x+342=199x^{2}+467x+378$$

Subtract 199x^{2} from both sides.

$$199x^{2}+459x+342-199x^{2}=467x+378$$

Combine 199x^{2} and -199x^{2} to get 0.

$$459x+342=467x+378$$

Subtract 467x from both sides.

$$459x+342-467x=378$$

Combine 459x and -467x to get -8x.

$$-8x+342=378$$

Subtract 342 from both sides.

$$-8x=378-342$$

Subtract 342 from 378 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.

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