To solve an equation in which the variable is within absolute value brackets, you must divide the equation into two equations. The most basic example of this is an equation of the form |x| = c. In this case, either x = c or x = –c. A slightly more complicated example is this:
In this problem, you must solve two equations: First, solve for x in the equation x + 3 = 5. In this case, x = 2. Then, solve for x in the equation x + 3 = –5. In this case, x = –8. So the solutions to the equation |x + 3| = 5 are x = {–8, 2}. Generally speaking, to solve an equation in which the variable is within absolute value brackets, first isolate the expression within the absolute value brackets and then divide the equation into two. Keep one of these two equations the same, while in the other negate one side of the equation. In either case, the absolute value of the expression within brackets will be the same. This is why there are always two solutions to absolute value problems (unless the variable is equal to 0). Here is one more example:
Solve for x in terms of y in the equation 3= y2 – 1.
First, isolate the expression within the absolute value brackets: Then solve for the variable as if the expression within absolute value brackets were positive: Next, solve for the variable as if the expression within absolute value brackets were negative: The solution set for x is {y2 – 3, –y2 –1}.