Question

How do you solve `\frac { 2x + 3} { x ( x + 2) } = \frac { 7} { x + 2}`?

Answer 1

It must be `x` not equal `0` and `x` not equal `-2` then we get by cross multiplication `2x+3=7x` so `3=5x` or `x=3/5`

Explanation:

For `x=0` or `x=-2` we get zero in the denominators.

Answer 2

`x=3/5`

Explanation:

`(2x+3)/(x(x+2))=7/(x+2)`

`"distribute denominator of fraction on left side"`

`(2x+3)/(x^2+2x)=7/(x+2)`

`"multiply both sides by "(x^2+2x)(x+2)`

`rArr(2x+3)(x+2)=7(x^2+2x)`

`"distributing both sides"`

`rArr2x^2+7x+6=7x^2+14x`

`"express in "color(blue)"standard form";ax^2+bx+c=0`

`"subtract "2x^2+7x+6" from both sides"`

`rArr0=5x^2+7x-6`

`"factor the quadratic using the a-c method"`

`"the factors of the product "5xx-6=-30`

`"which sum to + 7 are + 10 and - 3"`

`"split the middle term using these factors"`

`5x^2+10x-3x-6=0larrcolor(blue)"factor by grouping"`

`color(red)(5x)(x+2)color(red)(-3)(x+2)=0`

`"take out the "color(blue)"common factor "(x+2)`

`rArr(x+2)(color(red)(5x-3))=0`

`"equate each factor to zero and solve for x"`

`x+2=0rArrx=-2`

`x!=-2" as this makes the equation undefined"`

`5x-3=0rArrx=3/5larrcolor(blue)"is the solution"`