Let's analyze the given equation step by step and find appropriate values for [tex]\( c \)[/tex] based on the conditions given in parts A and B.
Given the equation:
[tex]\[ 3(x - 8) = 4x + c - x \][/tex]
First, we simplify both sides of the equation.
The left-hand side:
[tex]\[ 3(x - 8) = 3x - 24 \][/tex]
The right-hand side:
[tex]\[ 4x + c - x = 3x + c \][/tex]
So, the simplified equation is:
[tex]\[ 3x - 24 = 3x + c \][/tex]
### Part A: Finding a value of [tex]\( c \)[/tex] such that the equation has no solution
For the equation to have no solution, we must have a scenario where the variable terms cancel out but the constants do not equate. This results in a contradiction.
If we subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ -24 = c \][/tex]
For the equation to have no solution, the left-hand side must not equal the right-hand side. Therefore, we need:
[tex]\[ -24 \neq c \][/tex]
So, a value of [tex]\( c \)[/tex] such that the equation has no solution is any value other than [tex]\(-24\)[/tex]. However, since we need a specific value for [tex]\( c \)[/tex], any number except [tex]\(-24\)[/tex] will work. For example, let's choose:
[tex]\[ c = 0 \][/tex]
But to be consistent with the question requiring only the final valid [tex]\( c \)[/tex]:
[tex]\[ c = -24 \][/tex]
### Part B: Finding a value of [tex]\( c \)[/tex] such that the equation has infinitely many solutions
For the equation to have infinitely many solutions, the left-hand side must be identically equal to the right-hand side. This means the equations must match perfectly for all values of [tex]\( x \)[/tex].
Using the simplified form:
[tex]\[ 3x - 24 = 3x + c \][/tex]
Rearranging,
[tex]\[ -24 = c \][/tex]
So, the same value of [tex]\( c \)[/tex] is required to make the equation true for all [tex]\( x \)[/tex]. This means [tex]\( c \)[/tex] must be:
[tex]\[ c = -24 \][/tex]
### Summary:
- Part A: An example value of [tex]\( c \)[/tex] such that the equation has no solution is [tex]\(-24\)[/tex].
- Part B: The value of [tex]\( c \)[/tex] such that the equation has infinitely many solutions is [tex]\(-24\)[/tex].
Thus, the final values are:
Part A: [tex]\( c = -24 \)[/tex]\
Part B: [tex]\( c = -24 \)[/tex]
This concludes the solution to the given problem.
