solve the absolute value equation
|1+6c|-7=-3
To solve the absolute value equation |1+6c| - 7 = -3, we need to isolate the absolute value expression by moving constants and terms over to one side of the equation.
Step 1: First, let's add 7 to both sides of the equation to get rid of the -7 on the left side:
|1+6c| - 7 + 7 = -3 + 7
Simplifying the equation:
|1+6c| = 4
Step 2: We know that the absolute value of a number is always non-negative, so we can have two cases to solve the equation: when 1+6c is positive and when it is negative.
Case 1: 1+6c is positive, meaning 1+6c > 0.
In this case, the equation |1+6c| = 4 simplifies to:
1+6c = 4
Solving for c in this case:
1+6c = 4
Subtract 1 from both sides:
6c = 3
Divide both sides by 6:
c = 3/6
Simplifying the fraction:
c = 1/2
Case 2: 1+6c is negative, meaning 1+6c < 0.
In this case, we need to consider the absolute value of a negative number as its positive counterpart. So, the equation |1+6c| = 4 simplifies to:
-(1+6c) = 4
Solving for c in this case:
-(1+6c) = 4
Distribute the negative sign:
-1 - 6c = 4
Add 1 to both sides:
-6c = 5
Divide both sides by -6 (Remember to flip the inequality because we are dividing by a negative number):
c = 5/(-6)
Simplifying the fraction:
c = -5/6
So, the solutions to the absolute value equation |1+6c| - 7 = -3 are:
c = 1/2 and c = -5/6.