Inequalities, Number Line Graphs, and Interval Notation

Scope and Objectives of Unit Two: Inequalities and Absolute Value

• Unit two focuses on inequalities and absolute value.
• The foundational curriculum begins with mastering three distinct mathematical notation systems:
  • Graphical representation on a real number line.
  • Symbolic notation using algebraic inequality operators.
  • Interval notation utilizing boundary brackets and parentheses.
• The structural sequence examines specific number line graphs and their direct translations across notations, followed by a comprehensive synthesis of notation principles.

Coordinate Axis and Graphical Notation Fundamentals

• The real number line serves as the visual coordinate space for graphing inequality solutions:
  • The left terminus of the horizontal number line extends infinitely to negative infinity ($-\infty$).
  • The right terminus of the horizontal number line extends infinitely to positive infinity ($\infty$).
• Partition boundaries are marked at designated real number coordinates on the axis.
• The shaded region on a number line represents the continuous set of all numbers satisfying the condition.
• The variable $X$ is defined to represent all numbers contained within the shaded region.

Graph 1: Solid Endpoint and Rightward Shading

• Boundary configuration:
  • The critical boundary point on the number line is positioned at the value $3$.
  • The endpoint at coordinate $3$ is marked with a solid dot.
• Direction of shading:
  • The region is shaded continuously to the right, advancing from $3$ toward positive infinity ($\infty$).
• Mathematical meaning:
  • Rightward shading signifies values greater than the reference point.
  • A solid dot denotes inclusion of the boundary value $3$ within the solution set.
• Symbolic notation:
  • Inclusion requires an inequality symbol that incorporates equality.
  • Shading to the right requires the greater-than orientation.
  • The symbolic statement is written as:     $X \ge 3$
• Interval notation:
  • A square bracket is utilized to represent the inclusion of the endpoint $3$.
  • Unbounded expansion toward the right terminates at positive infinity ($\infty$).
  • The interval notation is written as:     $[3, \infty)$

Graph 2: Open Endpoint and Leftward Shading

• Boundary configuration:
  • The boundary point is situated at the real number coordinate $3$.
  • The endpoint at coordinate $3$ is depicted as an open circle.
• Direction of shading:
  • Shading covers all numbers extending to the left, progressing away from $3$ toward negative infinity ($-\infty$).
• Mathematical meaning:
  • Leftward shading indicates all real numbers strictly less than the boundary value $3$.
  • An open circle indicates that the number $3$ itself is excluded from the solution set.
• Symbolic notation:
  • The excluded boundary and leftward orientation specify a strict inequality without an equal sign:     $X < 3$
• Interval notation:
  • The interval spans from negative infinity ($-\infty$) on the left up to the upper boundary of $3$.
  • Exclusion of the endpoint requires the use of an open parenthesis rather than a square bracket:     $(-\infty, 3)$

Notation Conversion Rules and Summary

• Endpoint rules for boundary inclusion versus exclusion:
  • Solid dot on a graph indicates that the boundary value is included in the set; corresponds to an equal sign in symbolic notation ($\le$ or $\ge$) and a square bracket in interval notation ($[$ or $]$).
  • Open circle on a graph indicates that the boundary value is excluded from the set; corresponds to a strict inequality in symbolic notation ($<$ or $>$) and a round parenthesis in interval notation ($($ or $)$).
• Shading direction and inequality orientation:
  • Shading to the right corresponds to values greater than the boundary, bounded above by positive infinity ($\infty$).
  • Shading to the left corresponds to values less than the boundary, bounded below by negative infinity ($-\infty$).
• Infinity convention in interval notation:
  • Infinity ($\infty$) and negative infinity ($-\infty$) represent unbounded behavior rather than discrete numbers, and are always enclosed with parentheses ($($ or $)$), never square brackets.