SOLUTION: Hi, I have the question 5n + 24 = |8 - 3n| for my homework and I've gotten -2 and -16 as the answers, but I don't know if that's correct or if it would only be -2 as the answer

1.  First of all, you can easily check if your numbers satisfy the equation.

    For n = -2 you have

    a)  left side is  5*(-2) + 24 = -10 + 24 = 14.
    b)  right side is  |8 - 3*(*(-2)| = |8 + 6| = 14.

    So, n = -2 is the solution.



    For n = -16 you have 

    a)  left side is  5*(-16) + 24 = -80 + 24 = -56.
    b)  right side is  |8 - 3*(-16)| = |8 + 48| = 56.

    So, n = 16 is not the solution.



2.  The plots of the functions are shown below:


    


Plots y = 5x + 24 (red) and  y = |8 - 3x| (green)

3.  Now I will show you how to solve this equation

    5n + 24 = |8 - 3n|.     (1)


    Looking into the right sde, you can notice that there is the critical point n =  =  .


     Case a)  If  n <= ,  then 8 - 3n >= 0, and the right side of the equation (1) is 8-3n.

              Hence, in this case the equation (1) takes the form

              5n + 24 = 8 - 3n.    (2)

              Simplify it further and solve for n:

              5n + 3n = 8 - 24  ====>  8n = -16  ====>  n = -2.

              This value of n satisfy the inequality  n <= .
              Hence,  n = -2  is the solution to (2)  and  is the solution to (1).

              Case a)  is completed.



     Case b)  If  n > ,  then 8 - 3n < 0, and the right side of the equation (1) is 3n-8.

              Hence, in this case the equation (1) takes the form

              5n + 24 = 3n - 8.    (3)

              Simplify it further and solve for n:

              5n - 3n = -8 - 24  ====>  2n = -32  ====>  n = -16.

              But this value of n DOES NOT satisfy the inequality  n > .
              Hence,  n = -16, alghough  is the solution to (3),  IS NOT the solution to (1).

              Case b)  is completed.