Linear Inequalities and Graphing

Linear Inequalities

• Inequalities are relations between two values that are different.
• Inequality notations include >, <, ≥, ≤.
• Inequalities have properties like symmetric, transitive, additive, and multiplicative.
• A linear inequality can be written in the form Ax + B > 0, Ax + B < 0, Ax + B ≥ 0, or Ax + B ≤ 0, where A and B are real numbers.
• Solving linear inequalities is similar to solving linear equations, except the equal sign is replaced by the inequality symbols.

Graphing Linear Inequalities

• Graphing inequalities on a number line:
  • Use open circles (°) for > or <
  • Use closed circles (•) for ≥ or ≤
  • The direction of the arrow on the graph indicates the solution set. For example, x > 2 means all values greater than 2.
• Graphing inequalities incorporates using parenthesis and brackets.
  • If the inequality uses > or <, the parenthesis should be used. The parenthesis are used to show open interval or not including the exact number.
  • If the inequality uses ≥ or ≤, the bracket should be used. The bracket is used to show closed interval or including the exact number.

Compound Inequalities

• Compound inequalities combine two or more inequalities using "and" or "or".
  • "And"—both inequalities must be true simultaneously. The solution is where the graphs of both inequalities overlap
  • "Or"—at least one of the inequalities must be true. The solution is the union of the graphs of both inequalities.
• Solving compound inequalities involves solving each individual inequality and finding the intersection or union depending on whether the compound inequality uses "and" or "or".
• Solution sets for inequalities often involve infinity as part of the solution (-∞ or ∞).
• Solution sets are written using interval notation or parentheses to represent open intervals and brackets to represent closed intervals. For example, (-∞, 4] indicates all values from negative infinity up to and including 4.

This quiz explores the fundamentals of linear inequalities, including their notations, properties, and methods for solving them. It also covers how to graph these inequalities on a number line, emphasizing the use of open and closed circles to represent solutions. Test your understanding of both theory and application in this engaging quiz.