Question

Hello, I need help solving this problem. Thank you so much

Answer 1

$\frac{x}{x+4}+\frac{3}{x+5}=\frac{x+2}{x^2+9x+20}$

1. Factor the denominator of the expression on the irhgt of the equation:

- Write 9x as the sum of two terms that when you muliply the coefficient the result is 20:

$x^2+9x+20=x^2+5x+4x+20$

-Factor by parts: (common factor in frist two terms x and common factor in last two terms 4)

$=x(x+5)+4(x+5)$

-Factor (x+5)

$=(x+5)(x+4)$

Then, the given equation after the frist step is:

$\frac{x}{x+4}+\frac{3}{x+5}=\frac{x+2}{(x+5)(x+4)}$

2. Subtract the fraction on the right in both sides of the equation:

$\begin{gathered} \frac{x}{x+4}+\frac{3}{x+5}-\frac{x+2}{(x+5)(x+4)}=\frac{x+2}{(x+5)(x+4)}-\frac{x+2}{(x+5)(x+4)} \\ \\ \frac{x}{x+4}+\frac{3}{x+5}-\frac{x+2}{(x+5)(x+4)}=0 \end{gathered}$

3. Write each fraction with LCD (x+5)(x+4)

-First fraction: multiply numerator and denominator by (x+5):

$\frac{x}{x+4}\cdot\frac{x+5}{x+5}=\frac{x(x+5)}{(x+5)(x+4)}$

-Second fraction: multiply numerator and denominator by (x+4)

$\frac{3}{x+5}\cdot\frac{x+4}{x+4}=\frac{3(x+4)}{(x+5)(x+4)}$

-Third fraction is written with the LCD.

Then, the expression written with LCD is:

$\frac{x(x+5)}{(x+5)(x+4)}+\frac{3(x+4)}{(x+5)(x+4)}-\frac{x+2}{(x+5)(x+4)}=0$

4. Solve the operartions of the fractions:

$\frac{x(x+5)+3(x+4)-(x+2)}{(x+5)(x+4)}=0$

Simplify:

$\begin{gathered} \frac{x^2+5x+3x+12-x-2}{(x+5)(x+4)}=0 \\ \\ \frac{x^2+7x+10}{(x+5)(x+4)}=0 \end{gathered}$

5. Factor the numerator:

-Write 7x as the sum of two terms that when you muliply the coefficient the result is 10:

$x^2+7x+10=x^2+5x+2x+10$

-Factor by parts: (comon term in frist two terms is x, and comon factor in last two temrs is 2):

$=x(x+5)+2(x+5)$

-Factor (x+5):

$=(x+5)(x+2)$

Then, the equation after this step is:

$\frac{(x+5)(x+2)}{(x+5)(x+4)}=0$

6. Simplify:

$\frac{x+2}{x+4}=0$

7. Solve x:

When the quotient (result of division) is equal to 0, the numerator is equal to 0:

$\begin{gathered} x+2=0 \\ \\ \text{Subtract 2 in both sides of the equation:} \\ x+2-2=0-2 \\ x=-2 \end{gathered}$

Then, the solution for the given eqution is: x= -2