The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. |4x - 3| = |4x - 5|

Key Concepts

Absolute Value Definition

The absolute value of a number is its distance from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x|, which equals x if x is non-negative and -x if x is negative. This concept is crucial for understanding how to manipulate equations involving absolute values.

Equivalence of Absolute Value Equations

The equation |u| = |v| implies two possible scenarios: u = v or u = -v. This means that the expressions inside the absolute value can either be equal or opposites of each other. Recognizing this equivalence is essential for solving equations that contain absolute values, as it allows for the formulation of multiple equations to find solutions.

Solving Absolute Value Equations

To solve an absolute value equation, one must set up separate equations based on the equivalence of absolute values. For example, from |4x - 3| = |4x - 5|, we derive two equations: 4x - 3 = 4x - 5 and 4x - 3 = -(4x - 5). Solving these equations will yield the values of x that satisfy the original equation, highlighting the importance of systematic approaches in algebra.