Absolute Value Equations

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted as |x|.

To solve |x| = a, you set up two equations: x = a and x = -a.

The first step is to isolate the absolute value expression on one side of the equation.

It implies that the expression inside the absolute value can equal either 4x - 1 or -(4x - 1), leading to two separate equations to solve.

Substitute the solutions back into the original equation to verify if both sides are equal.

The possible solutions are x = 8 and x = -8.

This equation indicates that the expression inside the absolute value must be equal to 5x + 10 or its negative, leading to two equations.

If solving leads to a contradiction (e.g., a false statement), it means there are no solutions.

The solution set is x = 2.5 and x = -1.5.

Isolate the absolute value, then set up two equations: ax + b = c and ax + b = -c.