Final answer:
The sum of all solutions for x in the equation x² – 8x + 21 = |x – 4|+ 5 is infinite.
Step-by-step explanation:
To find the sum of all solutions for x in the equation, we can start by simplifying the right side of the equation. Since the absolute value function |x - 4| can be written as x - 4 or -(x - 4), we have two cases to consider:
Case 1: x - 4 = |x - 4|
Simplifying this equation gives x - 4 = x - 4. Since both sides are equal, any value of x will satisfy this equation.
Case 2: x - 4 = -(x - 4)
Simplifying this equation gives x - 4 = -x + 4. Adding x to both sides gives 2x - 4 = 4, and adding 4 to both sides gives 2x = 8. Dividing both sides by 2 gives x = 4, which is the only solution for this case.
Therefore, we have two solutions for x: x = 4 and x can be any value. The sum of all solutions is 4 + ∞ = ∞, which means there is no finite sum of solutions.