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. |x^2 - 6| = |5x|

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, as it allows us to consider both positive and negative scenarios.

Equivalence of Absolute Value Equations

The equation |u| = |v| implies two possible cases: u = v or u = -v. This property is essential when solving equations with absolute values, as it allows us to break down the problem into simpler linear equations. By considering both cases, we can find all possible solutions that satisfy the original equation.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. In the context of absolute value equations, once the cases are established, we can isolate the variable and use algebraic techniques such as addition, subtraction, multiplication, and division to solve for x. Mastery of these techniques is necessary to effectively tackle the equations derived from absolute value expressions.