select the correct answer.\nit would take charlene and gina 18 minutes to organize charlenes comic book…

select the correct answer.\nit would take charlene and gina 18 minutes to organize charlenes comic book collection if they work together. if gina works alone, it would take her 15 minutes longer than it would take charlene working alone.\nwhen x represents the number of minutes it takes charlene to organize the comic book collection working alone, the situation is represented by this equation:\n\\( \\frac { 1 } { x } + \\frac { 1 } { x + 15 } = \\frac { 1 } { 18 } \\).\nwhich value is a solution of the rational equation that must be discarded because of the context?\na. \\( x = 0 \\)\nb. \\( x = - 9 \\)\nc. \\( x = - 15 \\)\nd. \\( x = - 30 \\)

select the correct answer.\nit would take charlene and gina 18 minutes to organize charlenes comic book collection if they work together. if gina works alone, it would take her 15 minutes longer than it would take charlene working alone.\nwhen x represents the number of minutes it takes charlene to organize the comic book collection working alone, the situation is represented by this equation:\n\\( \\frac { 1 } { x } + \\frac { 1 } { x + 15 } = \\frac { 1 } { 18 } \\).\nwhich value is a solution of the rational equation that must be discarded because of the context?\na. \\( x = 0 \\)\nb. \\( x = - 9 \\)\nc. \\( x = - 15 \\)\nd. \\( x = - 30 \\)

Answer

Explanation:

Step1: Analyze the domain of the equation

The denominators in the equation (\frac{1}{x}+\frac{1}{x + 15}=\frac{1}{18}) cannot be zero. So (x\neq0) and (x+15\neq0) (i.e., (x\neq - 15)).

Step2: Check each option

  • Option A: If (x = 0), the first term (\frac{1}{x}) is undefined.
  • Option B: If (x=-9), (\frac{1}{-9}+\frac{1}{-9 + 15}=\frac{1}{-9}+\frac{1}{6}=\frac{-2 + 3}{18}=\frac{1}{18}), but we need to consider the context. Since (x) represents the number of minutes, a negative value does not make sense in the context of time.
  • Option C: If (x=-15), the second term (\frac{1}{x + 15}) is undefined.
  • Option D: If (x=-30), (\frac{1}{-30}+\frac{1}{-30+15}=\frac{1}{-30}+\frac{1}{-15}=\frac{-1-2}{30}=-\frac{1}{10}\neq\frac{1}{18}), and also negative value for time is not valid.

Since (x) represents time (number of minutes), negative values ((x=-9,x = - 15,x=-30)) do not make sense in this context. The value (x = 0) makes the equation undefined. But among the options, the reason for discarding based on context (time cannot be negative or zero in this work - rate problem where (x) is the time taken by Charlene) the most relevant is that (x=-9) gives a negative time value which is not practical for the number of minutes it takes to complete a task.

Answer:

B. (x = - 9)