SOLUTION: How do I solve: The absolute value of n-6 = the absolute value of 1-n

Question 84531: How do I solve:
The absolute value of n-6 = the absolute value of 1-n
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Given:
.

.
A methodical way of solving this equation involves understanding that each of the quantities
in the absolute value signs can be either plus or minus without changing the equation
... and this is due to the absolute value signs which change negative quantities to
positive quantities.
.
Therefore, you have 4 possibilities:
.
+(n - 6) = +(1 - n) and
-(n - 6) = +(1 - n) and
+(n - 6) = -(1 - n) and finally
-(n - 6) = -(1 - n)
.
Solve each of these four possible cases.
.
Case I.
.
+(n - 6) = +(1 - n) which is
n - 6 = 1 - n and add n to both sides to get
2n - 6 = 1 and add +6 to both sides to get
2n = 7 and divide by 2 to find that
n = 7/2
.
Case II
-(n - 6) = +(1 - n) and the minus sign changes the left side
-n + 6 = 1 - n now add n to both sides and you get
+6 = +1 ... this case does not lead to a good solution for n
.
Case III
.
+(n - 6) = -(1 - n) and the minus sign changes the right side
n - 6 = -1 + n subtract n from both sides and you get
-6 = -1 ... this case also does not lead to a good solution for n
.
Case IV
.
-(n - 6) = -(1 - n) and the minus signs on both sides change this to
-n + 6 = -1 + n subtract n from both sides and you get
-2n + 6 = -1 subtract 6 from both sides and you get
-2n = -7 divide both sides by -2 and the answer becomes
n = -7/-2 = 7/2 This is the same answer as Case I
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So the answer to the problem is n = +7/2. Let's check that by replacing n by 7/2 in the
equation originally given:
.

.
Substitute 7/2 for n and this equation becomes:
.

.
Make each of the terms inside the absolute value signs have a common denominator of 2:
.

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Now combine the terms inside the absolute value signs and the equation becomes:
.

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Both sides of this equation are the same, so we can say that n = 7/2 is a solution
to the equation.
.
Hope this helps you understand absolute values a little more.
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