Question

Which of the following shows the correct simplification assuming the denominator is not equal to zero?

Answer 1

`=(3x+2)/(x\left(x+1\right))`

or

`((3x+2))/(x^2+x)`

Explanation:

`(2)/(x)+(1)/(x+1)\\\mathrm{Least\:Common\:Multiplier\:of\:}x,\:x+1:\quad x\left(x+1\right)\\Adjust\:Fractions\:based\:on\:the\:LCM\\Multiply\:each\:numerator\:by\:the\:same\:amount\:needed\:to\:multiply\:its\\\mathrm{corresponding\:denominator\:to\:turn\:it\:into\:the\:LCM}\:x\left(x+1\right)\\\mathrm{For}\:(2)/(x):\:\mathrm{multiply\:the\:denominator\:and\:numerator\:by\:}\:x+1\\(2)/(x)=(2\left(x+1\right))/(x\left(x+1\right))\\`

`\mathrm{For}\:(1)/(x+1):\:\mathrm{multiply\:the\:denominator\:and\:numerator\:by\:}\:x\\(1)/(x+1)=(1\cdot \:x)/(\left(x+1\right)x)=(x)/(x\left(x+1\right))\\=(2\left(x+1\right))/(x\left(x+1\right))+(x)/(x\left(x+1\right))\\\mathrm{Since\:the\:denominators\:are\:equal,\:combine\:the\:fractions}:\quad (a)/(c)\pm (b)/(c)=(a\pm \:b)/(c)\\=(2\left(x+1\right)+x)/(x\left(x+1\right))\\`

`\mathrm{Expand}\:2\left(x+1\right)+x:\quad 3x+2\\2\left(x+1\right)+x\\\mathrm{Apply\:the\:distributive\:law}:\quad \:a\left(b+c\right)=ab+ac\\a=2,\:b=x,\:c=1\\=2x+2\cdot \:1\\\mathrm{Multiply\:the\:numbers:}\:2\cdot \:1=2\\=2x+2\\\mathrm{Simplify}\:2x+2+x:\quad 3x+2\\2x+2+x\\\mathrm{Group\:like\:terms}\\=2x+x+2\\\\mathrm{Add\:similar\:elements:}\:2x+x=3x\\=3x+2\\=(3x+2)/(x\left(x+1\right))`