Question

Hello, I need help solving this practice problems with showing appropriate work for each step. Thank you!

Answer 1

Computing the sum of fractions on the left side of the equation, we get:

`(x(x+5)+3(x+4))/((x+4)(x+5))=(x+2)/(x^2+9x+20).`

Now, notice that:

`x^2+9x+20=(x+4)(x+5).`

Therefore, we can rewrite the equation as:

`(x(x+5)+3(x+4))/((x+4)(x+5))=(x+2)/((x+4)(x+5)).`

The above result implies that:

`x(x+5)+3(x+4)=x+2.`

Simplifying the above equation, we get:

`\begin{gathered} x^2+5x+3x+12=x+2, \\ x^2+8x+12=x+2. \end{gathered}`

Subtracting x+2, we get:

`x^2+7x+10=0.`

Notice that:

`x^2+7x+10=(x+2)(x+5).`

Therefore,

`x=-2,\text{ }`

because x≠-5, otherwise we could not have a denominator of x+5.

Answer:

`x=2.`