Use the definition of absolute value to solve the following equation.
x-1/3=1/2x+1/6
To solve the given equation, we need to use the definition of absolute value to isolate the variable.
The definition of absolute value states that for any real number x, the absolute value of x (written as |x|) is equal to x if x is positive or zero, and it is equal to the negative of x if x is negative.
Now let's solve the equation step by step:
1. Start by isolating the absolute value expression. Move the term with x to one side of the equation by subtracting 1/2x from both sides and move the constant term to the other side by subtracting 1/3 from both sides:
x - (1/2)x = 1/6 + 1/3
Simplifying:
(1 - 1/2)x = 1/6 + 2/6
(1/2)x = 3/6
(1/2)x = 1/2
2. To remove the absolute value symbol, we need to consider two cases:
Case 1: |x| = x
In this case, the equation becomes:
(1/2)x = 1/2
3. Solve for x in case 1:
By multiplying both sides of the equation by 2, we find:
x = 1
Case 2: |x| = -x
In this case, the equation becomes:
(-1/2)x = 1/2
4. Solve for x in case 2:
By multiplying both sides of the equation by -2, we find:
x = -1
So, the equation has two possible solutions: x = 1 and x = -1.