simplify u^2+3u/u^2-9A.u/u-3, =/ -3, and u=/3B. u/u-3, u=/-3

  The correct answer is:  Answer choice:  [A]:
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→  "[tex] \frac{u}{u-3} [/tex] " ;  " { u [tex] \neq [/tex] ± 3 } " ; 

          →  or, write as:  " u / (u − 3) " ;  {" u ≠ 3 "}  AND:  {" u ≠ -3 "} ; 
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Explanation:
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 We are asked to simplify:
  
  [tex] \frac{(u^2+3u)}{(u^2-9)} [/tex] ;  


Note that the "numerator" —which is:  "(u² + 3u)" — can be factored into:
                                                      →  " u(u + 3) " ;

And that the "denominator" —which is:  "(u² − 9)" — can be factored into:
                                                      →   "(u − 3) (u + 3)" ;
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Let us rewrite as:
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→    [tex] \frac{u(u+3)}{(u-3)(u+3)} [/tex]  ;

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→  We can simplify by "canceling out" BOTH the "(u + 3)" values; in BOTH the "numerator" AND the "denominator" ;  since:

" [tex] \frac{(u+3)}{(u+3)} = 1 [/tex] "  ;

→  And we have:
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→  " [tex] \frac{u}{u-3} [/tex] " ;   that is:  " u / (u − 3) " ;  { u[tex] \neq 3[/tex] } .
                                                                                and:  { u[tex] \neq-3[/tex] } .

→ which is:  "Answer choice:  [A] " .
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NOTE:  The "denominator" cannot equal "0" ; since one cannot "divide by "0" ; 

and if the denominator is "(u − 3)" ;  the denominator equals "0" when "u = -3" ;  as such:

"u[tex] \neq [/tex]3" ; 

→ Note:  To solve:  "u + 3 = 0" ; 

 Subtract "3" from each side of the equation; 

                       →  " u + 3 − 3 = 0 − 3 " ; 

                       → u =  -3 (when the "denominator" equals "0") ; 
 
                       → As such:  " u [tex] \neq [/tex] -3 " ; 

Furthermore, consider the initial (unsimplified) given expression:

→  [tex] \frac{(u^2+3u)}{(u^2-9)} [/tex] ;  

Note:  The denominator is:  "(u²  − 9)" . 

The "denominator" cannot be "0" ; because one cannot "divide" by "0" ; 

As such, solve for values of "u" when the "denominator" equals "0" ; that is:
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→  " u² − 9 = 0 " ; 

 →  Add "9" to each side of the equation ; 

 →  u² − 9 + 9 = 0 + 9 ; 

 →  u² = 9 ; 

Take the square root of each side of the equation; 
 to isolate "u" on one side of the equation; & to solve for ALL VALUES of "u" ; 

→ √(u²) = √9 ; 

→ | u | = 3 ; 

→  " u = 3" ; AND;  "u = -3 " ; 

We already have:  "u = -3" (a value at which the "denominator equals "0") ; 

We now have "u = 3" ; as a value at which the "denominator equals "0"); 

→ As such: " u[tex] \neq 3[/tex]" ; "u [tex] \neq [/tex] -3 " ;  

or, write as:  " { u [tex] \neq [/tex] ± 3 } " .

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