Solve 1/x+5+1/x+8=1/x+6+1/x+7

$$\frac{ 1 }{ x+5 } + \frac{ 1 }{ x+8 } = \frac{ 1 }{ x+6 } + \frac{ 1 }{ x+7 }$$

Solve for x

$$x = -\frac{13}{2} = -6\frac{1}{2} = -6.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -8,-7,-6,-5 since division by zero is not defined. Multiply both sides of the equation by \left(x+5\right)\left(x+6\right)\left(x+7\right)\left(x+8\right), the least common multiple of x+5,x+8,x+6,x+7.

$$\left(x+6\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+7\right)=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+6 by x+7 and combine like terms.

$$\left(x^{2}+13x+42\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+7\right)=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x^{2}+13x+42 by x+8 and combine like terms.

$$x^{3}+21x^{2}+146x+336+\left(x+5\right)\left(x+6\right)\left(x+7\right)=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+5 by x+6 and combine like terms.

$$x^{3}+21x^{2}+146x+336+\left(x^{2}+11x+30\right)\left(x+7\right)=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x^{2}+11x+30 by x+7 and combine like terms.

$$x^{3}+21x^{2}+146x+336+x^{3}+18x^{2}+107x+210=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

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

$$2x^{3}+21x^{2}+146x+336+18x^{2}+107x+210=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Combine 21x^{2} and 18x^{2} to get 39x^{2}.

$$2x^{3}+39x^{2}+146x+336+107x+210=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Combine 146x and 107x to get 253x.

$$2x^{3}+39x^{2}+253x+336+210=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Add 336 and 210 to get 546.

$$2x^{3}+39x^{2}+253x+546=\left(x+5\right)\left(x+7\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+5 by x+7 and combine like terms.

$$2x^{3}+39x^{2}+253x+546=\left(x^{2}+12x+35\right)\left(x+8\right)+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x^{2}+12x+35 by x+8 and combine like terms.

$$2x^{3}+39x^{2}+253x+546=x^{3}+20x^{2}+131x+280+\left(x+5\right)\left(x+6\right)\left(x+8\right)$$

Use the distributive property to multiply x+5 by x+6 and combine like terms.

$$2x^{3}+39x^{2}+253x+546=x^{3}+20x^{2}+131x+280+\left(x^{2}+11x+30\right)\left(x+8\right)$$

Use the distributive property to multiply x^{2}+11x+30 by x+8 and combine like terms.

$$2x^{3}+39x^{2}+253x+546=x^{3}+20x^{2}+131x+280+x^{3}+19x^{2}+118x+240$$

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

$$2x^{3}+39x^{2}+253x+546=2x^{3}+20x^{2}+131x+280+19x^{2}+118x+240$$

Combine 20x^{2} and 19x^{2} to get 39x^{2}.

$$2x^{3}+39x^{2}+253x+546=2x^{3}+39x^{2}+131x+280+118x+240$$

Combine 131x and 118x to get 249x.

$$2x^{3}+39x^{2}+253x+546=2x^{3}+39x^{2}+249x+280+240$$

Add 280 and 240 to get 520.

$$2x^{3}+39x^{2}+253x+546=2x^{3}+39x^{2}+249x+520$$

Subtract 2x^{3} from both sides.

$$2x^{3}+39x^{2}+253x+546-2x^{3}=39x^{2}+249x+520$$

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

$$39x^{2}+253x+546=39x^{2}+249x+520$$

Subtract 39x^{2} from both sides.

$$39x^{2}+253x+546-39x^{2}=249x+520$$

Combine 39x^{2} and -39x^{2} to get 0.

$$253x+546=249x+520$$

Subtract 249x from both sides.

$$253x+546-249x=520$$

Combine 253x and -249x to get 4x.

$$4x+546=520$$

Subtract 546 from both sides.

$$4x=520-546$$

Subtract 546 from 520 to get -26.

$$4x=-26$$

Divide both sides by 4.

$$x=\frac{-26}{4}$$

Reduce the fraction \frac{-26}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{13}{2}$$