Page 2: Equations with Two Absolute Values

This page delves into more complex equations with absolute values, specifically those involving two absolute value expressions. It provides a comprehensive approach to solving such equations.

The page presents an example equation: |x| + |x - 1| = 2. The solution process involves considering multiple cases based on the possible combinations of positive and negative values inside each absolute value.

Example: For |x| + |x - 1| = 2, four cases are considered:

1. x ≥ 0 and x - 1 ≥ 0
2. x ≥ 0 and x - 1 < 0
3. x < 0 and x - 1 ≥ 0 (impossible case)
4. x < 0 and x - 1 < 0

The page introduces the concept of a sign table to systematically analyze these cases. This table helps in visualizing the intervals where each absolute value expression is positive or negative.

Highlight: When solving equations with two absolute values, it's crucial to consider all possible combinations of signs and to check the validity of each solution within its respective interval.

The page concludes with a complete solution, demonstrating how to combine the results from each case to obtain the final set of solutions.

Page 3: Equations with Nested Absolute Values

This page tackles the most advanced type of equations with absolute values: those with nested absolute values. It provides a detailed walkthrough of solving an equation where one absolute value expression is contained within another.

The example equation presented is ||x - 2| - x| = 3x. The solution process for this type of equation involves a systematic approach:

1. First, analyze the inner absolute value expression.
2. Then, consider the outer absolute value.
3. Set up multiple cases based on the possible sign combinations.

Example: For ||x - 2| - x| = 3x, the following cases are considered:

1. x - 2 < 0 and |x - 2| - x < 0
2. x - 2 ≥ 0 and x - $x - 2$ < 0
3. x - 2 ≥ 0 and x - $x - 2$ ≥ 0

The page emphasizes the importance of carefully analyzing each case and checking the validity of solutions within their respective intervals.

Highlight: When solving equations with nested absolute values, it's crucial to work from the innermost absolute value outward, considering all possible sign combinations at each level.

The document concludes with a comprehensive solution, demonstrating how to combine the results from each case to obtain the final set of solutions for this complex equation.

Page 1: Fundamental Properties and Single Absolute Value Equations

This page introduces the basic concepts of equations with absolute values and provides a detailed explanation of solving equations with a single absolute value.

Definition: The absolute value of a number is its distance from zero on a number line, regardless of whether it's positive or negative.

The fundamental property of absolute values is presented, which states that |x| = x if x ≥ 0, and |x| = -x if x < 0. This property is crucial for solving equations with absolute values.

Example: For the equation |x + 1| = 2, two cases are considered:

1. When x + 1 ≥ 0: x + 1 = 2, solving to x = 1
2. When x + 1 < 0: -$x + 1$ = 2, solving to x = -3

The page emphasizes the importance of considering both positive and negative cases when solving equations with absolute values. It also provides a step-by-step approach to solving these equations, which includes identifying the expression inside the absolute value, setting up two equations based on the possible signs, and solving each equation separately.

Highlight: Always check if the solutions satisfy the original equation and the conditions set for each case.