Solve 1/x+4+1/x+6=1/x+7+1/x+3

$$\frac{ 1 }{ x+4 } + \frac{ 1 }{ x+6 } = \frac{ 1 }{ x+7 } + \frac{ 1 }{ x+3 }$$

Solve for x

$$x=-5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -7,-6,-4,-3 since division by zero is not defined. Multiply both sides of the equation by \left(x+3\right)\left(x+4\right)\left(x+6\right)\left(x+7\right), the least common multiple of x+4,x+6,x+7,x+3.

$$\left(x+3\right)\left(x+6\right)\left(x+7\right)+\left(x+3\right)\left(x+4\right)\left(x+7\right)=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

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

$$\left(x^{2}+9x+18\right)\left(x+7\right)+\left(x+3\right)\left(x+4\right)\left(x+7\right)=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

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

$$x^{3}+16x^{2}+81x+126+\left(x+3\right)\left(x+4\right)\left(x+7\right)=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

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

$$x^{3}+16x^{2}+81x+126+\left(x^{2}+7x+12\right)\left(x+7\right)=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

Use the distributive property to multiply x^{2}+7x+12 by x+7 and combine like terms.

$$x^{3}+16x^{2}+81x+126+x^{3}+14x^{2}+61x+84=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

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

$$2x^{3}+16x^{2}+81x+126+14x^{2}+61x+84=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

Combine 16x^{2} and 14x^{2} to get 30x^{2}.

$$2x^{3}+30x^{2}+81x+126+61x+84=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

Combine 81x and 61x to get 142x.

$$2x^{3}+30x^{2}+142x+126+84=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

Add 126 and 84 to get 210.

$$2x^{3}+30x^{2}+142x+210=\left(x+3\right)\left(x+4\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

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

$$2x^{3}+30x^{2}+142x+210=\left(x^{2}+7x+12\right)\left(x+6\right)+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

Use the distributive property to multiply x^{2}+7x+12 by x+6 and combine like terms.

$$2x^{3}+30x^{2}+142x+210=x^{3}+13x^{2}+54x+72+\left(x+4\right)\left(x+6\right)\left(x+7\right)$$

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

$$2x^{3}+30x^{2}+142x+210=x^{3}+13x^{2}+54x+72+\left(x^{2}+10x+24\right)\left(x+7\right)$$

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

$$2x^{3}+30x^{2}+142x+210=x^{3}+13x^{2}+54x+72+x^{3}+17x^{2}+94x+168$$

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

$$2x^{3}+30x^{2}+142x+210=2x^{3}+13x^{2}+54x+72+17x^{2}+94x+168$$

Combine 13x^{2} and 17x^{2} to get 30x^{2}.

$$2x^{3}+30x^{2}+142x+210=2x^{3}+30x^{2}+54x+72+94x+168$$

Combine 54x and 94x to get 148x.

$$2x^{3}+30x^{2}+142x+210=2x^{3}+30x^{2}+148x+72+168$$

Add 72 and 168 to get 240.

$$2x^{3}+30x^{2}+142x+210=2x^{3}+30x^{2}+148x+240$$

Subtract 2x^{3} from both sides.

$$2x^{3}+30x^{2}+142x+210-2x^{3}=30x^{2}+148x+240$$

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

$$30x^{2}+142x+210=30x^{2}+148x+240$$

Subtract 30x^{2} from both sides.

$$30x^{2}+142x+210-30x^{2}=148x+240$$

Combine 30x^{2} and -30x^{2} to get 0.

$$142x+210=148x+240$$

Subtract 148x from both sides.

$$142x+210-148x=240$$

Combine 142x and -148x to get -6x.

$$-6x+210=240$$

Subtract 210 from both sides.

$$-6x=240-210$$

Subtract 210 from 240 to get 30.

$$-6x=30$$

Divide both sides by -6.

$$x=\frac{30}{-6}$$

Divide 30 by -6 to get -5.

$$x=-5$$