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. |3x - 1| = |x + 5|

Key Concepts

Absolute Value

Absolute value measures the distance of a number 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: either u = v or u = -v. This property allows us to break down absolute value equations into simpler linear equations, making it easier to find solutions. Recognizing this equivalence is essential for solving equations that contain absolute values.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This process often includes isolating the variable on one side of the equation through algebraic manipulation. Understanding how to solve linear equations is fundamental when working with the results obtained from absolute value equations.