This page delves into the concept of absolute value equations and inequalities, providing comprehensive explanations and multiple solution methods.

Definition: Absolute value represents the distance from zero on a number line, always resulting in a positive value or zero.

Example: In solving |x + 5| = 2, two cases must be considered:

1. Positive case: x + 5 = 2
2. Negative case: x + 5 = -2

Highlight: When solving absolute value equations, always consider both positive and negative possibilities to find all valid solutions.

Example: For 2|x + 5| = 4:

1. Simplify to |x + 5| = 2
2. Solve for x + 5 = 2 and x + 5 = -2
3. Solutions are x = -3 and x = -7

Vocabulary: The term "no solution" indicates when an absolute value equation has no real number solutions that satisfy the given conditions.

This page introduces the fundamental concepts of compound inequalities and their solutions. The content focuses on two main types of compound inequalities: 'and' and 'or' operations, with detailed explanations of how to solve them.

Definition: Compound inequalities are mathematical statements that combine two or more inequalities using 'and' or 'or' operations.

Example: For the inequality -3≤x≤3, x must be greater than or equal to -3 AND less than or equal to 3.

Highlight: When solving compound inequalities with 'and', the solution must satisfy both conditions simultaneously.

The page demonstrates step-by-step solutions for various compound inequalities:

Example: Solving -3x+2>-7:

1. Add 2 to both sides: -3x>-9
2. Divide by -3: x<3

Vocabulary: The term "middle" refers to the value of x that must fall between two boundary points in an 'and' compound inequality.