Solve the equation.
|n – 6| = |1 – n|
2 choices:
n-6 = 1-n or -n + 6 = 1-n
1.
n-6 = 1-n
2n = 7
n = 7/2
2.
-n+6 = 1-n
6 =- 1, contradiction, so no solution
so n = 7/2
To solve the equation |n – 6| = |1 – n|, we will consider the case where each absolute value expression is positive and the case where each absolute value expression is negative.
Case 1: both expressions are positive:
When both expressions inside the absolute value bars are positive, you can remove the absolute value bars without changing the equation.
For the left side, n - 6 is positive:
n - 6 > 0
Adding 6 to both sides, we have:
n > 6
For the right side, 1 - n is positive:
1 - n > 0
Adding n to both sides and subtracting 1 from both sides, we have:
1 > n
Analyzing both inequalities, we see that n must be greater than 6 and less than 1. However, there are no numbers that satisfy this condition, so there are no solutions in this case.
Case 2: both expressions are negative:
When both expressions inside the absolute value bars are negative, we can again remove the absolute value bars without changing the equation.
For the left side, n - 6 is negative:
n - 6 < 0
Adding 6 to both sides, we have:
n < 6
For the right side, 1 - n is negative:
1 - n < 0
Adding n to both sides and subtracting 1 from both sides, we have:
1 < n
Analyzing both inequalities, we see that n must be less than 6 and greater than 1. Again, there are no numbers satisfying this condition, so there are no solutions in this case either.
Thus, there are no solutions to the equation |n – 6| = |1 – n|.