Question

Please explain how to solve one absolute value equation by writing it as two equations.

Answer 1

Well we know that an absolute value will always return an answer that is nonnegative.

|x| = x if x >= 0, -x if x < 0

For example, y = |5x - 3|. If 5x - 3 >= 0 then 5x >= 3, and x >= 3/5. So whenever x >= 3/5, then y = 5x - 3. Otherwise y = -(5x -3).

You can solve these by setting up two equations. One equation equal to whatever is inside the absolute value function. And another equation equal to the negative of that.

If y = |5x - 3| then

y = 5x - 3 or y = -5x + 3.

A working example:

|3x - 5| = 7

This means 3x - 5 = 7 or -(3x - 5) = 7

Solve each equation separately.

3x - 5 = 7
3x = 12
x = 4

-(3x - 5) = 7
3x - 5 = -7
3x = -2
x = -2/3

x = 4 or -2/3