Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | x2 + 1 | - | 2x | = 0

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. Understanding absolute value is crucial for solving equations and inequalities that involve expressions that can take on both positive and negative values.

Properties of Absolute Value

The properties of absolute value include the fact that |a| = |b| implies a = b or a = -b. This property allows us to set up two separate equations when dealing with absolute values in equations. Additionally, the property |a| - |b| = 0 indicates that the two absolute values must be equal, which is essential for solving the given equation.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. In the context of absolute value equations, this often requires isolating the absolute value expression and then applying the properties of absolute value to create separate cases. This method is essential for determining all possible solutions to the equation, especially when the equation involves multiple absolute value terms.