Question

Can someone please check my answer fast?

Answer 1

Your answer was:  "g+11 over/ 2x+15 " . 
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Your answer was "incorrect —but almost correct" !

Instead of "(g + 11)" for the "numerator" ; you should have put:  "(x + 11)" .

As a matter of technicality, you could have/should have stated:
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 {  $x \neq - 7.5$ } ; { $x \neq -2.5$ }. 
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    →   {but this would depend on the context — and/or the requirements of the course/instructor.}.  Good job!
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Explanation:
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Given:   g(x) =  $\frac{(x+6)}{(2x + 5)}$ ;

Find:  g(x+5) .
 
To do so, we plug in "(x+5)" for all values of "x" in the equation; & solve:
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        Start with the "numerator":  "(x + 6)" :

→  (x + 5 + 6) = x + 11 ; 
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Then, examine the "denominator" :  "(2x + 5)"

→ 2(x+5) + 5 ; 

   →  2(x + 5) = 2*x + 2*5 = 2x + 10 ;  


→ 2(x+5) + 5 = 

        2x + 10 + 5 ; 

    =  2x + 15 ; 
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→  g(x + 5) =  $\frac{x+11}{2x +15}$  . 

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Note that the "denominator" cannot equal "0" ;
         since one cannot "divide by "0" ; 
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So, given the denominator:  "2x + 15" ; 

→  at what value for "x" does  the denominator, "2x + 15" , equal "0" ?

→  2x + 15 = 0 ; 

Subtract "15" from each side of the equation:

→  2x + 15 - 15 = 0 - 15 ; 

to get: 

→  2x = -15 ; 

Divide EACH SIDE of the equation by "2" ; 
    To isolate "x" on one side of the equation; & to solve for "x" ; 

→  2x / 2  =  -15 / 2 ; 

to get: 

→  x = - 7. 5 ;  
Your answer was:  "g+11 over/ 2x+15 " . 
____________________________________________________
Your answer was "incorrect —but almost correct" !

Instead of "(g + 11)" for the "numerator" ; you should have put:  "(x + 11)" .

As a matter of technicality, you could have/should have stated:
________________________________________________________

 {  $x \neq - 7.5$ } ; { $x \neq -2.5$ }. 
________________________________________________________
    →   {but this would depend on the context — and/or the requirements of the course/instructor.}.  Good job!
________________________________________________________
So;  " $x \neq - 7.5$ " .
________________________________________________________
Now, examine the "denominator" from the original equation:
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→  "(2x + 5)"  ;  

→  At what value for "x" does the 'denominator' equal "0" ? 

→  2x + 5 = 0 ; 

Subtract "5" from each side of the equation: 

→  2x + 5 - 5 = 0 - 5 ; 

to get:

→  2x = -5 ; 

Divide each side of the equation by "2" ; 
     to isolate "x" on one side of the equation; & to solve for "x" ; 

→  2x / 2 = -5 / 2 ;

→  x = -2.5 ; 

→  So;  " $x \neq -2.5$ " .
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The correct answer is:
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 →  g(x + 5) =  $\frac{x+11}{2x +15}$ ;

         {  $x \neq - 7.5$ } ; { $x \neq -2.5$ }. 
____________________________________________________

→ Your answer was:  "g+11 over/ 2x+15 " . 
____________________________________________________
Your answer was "incorrect —but almost correct" !

Instead of "(g + 11)" for the "numerator" ; you should have put:  "(x + 11)" .

As a matter of technicality, you could have/should have stated:
________________________________________________________

 {  $x \neq - 7.5$ } ; { $x \neq -2.5$ }. 
________________________________________________________
    →   {but this would depend on the context — and/or the requirements of the course/instructor.}.  Good job!
________________________________________________________