Comprehensive Guide to Inequalities, Number Line Graphing, and Interval Notation

Fundamental Concepts of Inequalities

• An inequality is an algebraic statement expressing that one quantity is strictly greater than, greater than or equal to, strictly less than, or less than or equal to another quantity.
• Representation of strict inequalities:
  • The inequality $x > 4$ indicates that $x$ can be any real number strictly larger than $4$.
  • Strict inequality symbols ($>$ or $<$) do not include the boundary value in the solution set.
• Representation of non-strict inequalities:
  • The inequality $x \text{ } \backslash \text{le } 4$ includes the boundary value $4$ in the solution set due to the horizontal line at the bottom of the symbol ($\text{ } \backslash \text{le }$).
  • Non-strict inequality symbols ($\text{ } \backslash \text{le }$ or $\text{ } \backslash \text{ge }$) include the boundary value in the solution set.

Graphical Representations on the Number Line

• Number line representation of strict inequalities (e.g., $x > 4$):
  • Locate the boundary point $4$ on the number line.
  • Numbers larger than $4$ are located to the right of $4$, so the highlighting or arrow extends to the right.
  • Since $4$ is not included, it is represented visually on the number line in one of two valid ways:
  • Method A: An open / blank circle at $4$ with an arrow pointing to the right.
  • Method B: An open parenthesis ( at $4$ with an arrow pointing to the right.
  • Method Comparison:
  • The open circle method is preferred for general graphical visualization on standard number line graphs.
  • The parenthesis method directly corresponds to interval notation.
  • Both visual representations are valid and acceptable on examinations.
• Number line representation of non-strict inequalities (e.g., $x \text{ } \backslash \text{le } 4$):
  • Locate the boundary point $4$ on the number line.
  • Numbers less than $4$ are located to the left of $4$, so the arrow extends to the left.
  • Since $4$ is included in the solution set, it is represented on the number line in one of two valid ways:
  • Method A: A filled-in / solid circle at $4$ pointing to the left.
  • Method B: A square bracket ] at $4$ pointing to the left.
  • Square brackets directly correspond to inclusion in interval notation.

Interval Notation Rules and Conventions

• General Rules for Interval Notation:
  • Interval notation is always written reading from left to right across the number line (from the minimum value to the maximum value).
  • Exclusion: An open parenthesis ( or ) is used when a boundary value is not included in the set.
  • Inclusion: A square bracket [ or ] is used when a boundary value is included in the set.
• Rules for Infinity Symbols:
  • Unbounded sets extending to the far right proceed conceptually to positive infinity ($\text{ } \backslash \text{infty }$).
  • Unbounded sets extending to the far left proceed conceptually to negative infinity ($-\text{ } \backslash \text{infty }$).
  • Infinity ($\text{ } \backslash \text{infty }$) and negative infinity ($-\text{ } \backslash \text{infty }$) are concepts representing unbounded values rather than distinct real numbers, so they are never included in an interval.
  • Infinity symbols must always be paired with an open parenthesis ( or ).
• Interval Notation Examples:
  • For $x > 4$: The interval starts above $4$ (not included) and extends infinitely to the right, written as $(4, \text{ } \backslash \text{infty })$.
  • For $x \text{ } \backslash \text{le } 4$: The interval starts infinitely to the left and ends at $4$ (included), written as $(-\text{ } \backslash \text{infty }, 4]$.

Composite Inequalities with "AND" (Intersections)

• Definition of Composite / Compound Inequalities:
  • A composite inequality combines two separate inequality statements using logical connectives.
• Meaning of the Connective "AND":
  • An "AND" compound inequality requires finding the set of numbers that makes both individual inequality statements true simultaneously.
  • Graphically, this corresponds to the overlapping region where both inequality lines touch on a number line.
• Analysis of Example: $x > -7$ AND $x \text{ } \backslash \text{le } 3$
  • Graphing component 1 ($x > -7$): Starts at $-7$ (not included, open boundary) and extends infinitely to the right.
  • Graphing component 2 ($x \text{ } \backslash \text{le } 3$): Starts at $3$ (included, closed boundary) and extends infinitely to the left.
  • Finding the Overlap:
  • Left boundary: Stops at $-7$. The value $-7$ is not included because it does not satisfy $x > -7$ (it is not touched by both lines).
  • Right boundary: Stops at $3$. The value $3$ is included because the line for $x > -7$ continues past $3$ without gaps, and $3$ is explicitly included in $x \text{ } \backslash \text{le } 3$.
  • Resulting Interval Notation: $(-7, 3]$.
• Rewriting as a Single Bounded Inequality:
  • A compound "AND" inequality can be rewritten as a single combined inequality string.
  • Statement: $x > -7$ AND $x \text{ } \backslash \text{le } 3$ is logically equivalent to $-7 < x \text{ } \backslash \text{le } 3$.
  • Derivation: Since $x > -7$ is equivalent to $-7 < x$, joining it directly with $x \text{ } \backslash \text{le } 3$ gives $-7 < x \text{ } \backslash \text{le } 3$, placing $x$ directly between $-7$ and $3$.

Composite Inequalities with "OR" (Unions)

• Meaning of the Connective "OR":
  • An "OR" compound inequality requires that a given number $x$ satisfies at least one of the individual inequalities (or both).
  • An element is part of the solution set if it belongs to either set.
• Analysis of Example: $x \text{ } \backslash \text{le } -4$ OR $x > 2$
  • Testing Specific Sample Values:
  • Test $x = -5$: Satisfies $-5 \text{ } \backslash \text{le } -4$ (Valid solution).
  • Test $x = 4$: Satisfies $4 > 2$ (Valid solution, despite failing $4 \text{ } \backslash \text{le } -4$).
  • Test $x = 2.1$: Satisfies $2.1 > 2$ (Valid solution).
  • Test $x = 2$: Fails $2 \text{ } \backslash \text{le } -4$ and fails $2 > 2$ (Strict inequality excludes $2$; Invalid solution).
  • Graphing on the Number Line:
  • Left interval ray: Starts at $-4$ (solid circle or square bracket ]) pointing left towards negative infinity ($-\text{ } \backslash \text{infty }$).
  • Right interval ray: Starts at $2$ (open circle or parenthesis () pointing right towards positive infinity ($\text{ } \backslash \text{infty }$).
  • Results in two separate, disjoint rays on the number line.

Set Operations: Union versus Intersection

• The Union Symbol ($\text{ } \backslash \text{cup }$):
  • Representing "OR" statements in interval notation requires combining separate intervals using the union symbol $\text{ } \backslash \text{cup }$
  • Interval for left ray: $(-\text{ } \backslash \text{infty }, -4]$
  • Interval for right ray: $(2, \text{ } \backslash \text{infty })$
  • Complete combined representation: $(-\text{ } \backslash \text{infty }, -4] \text{ } \backslash \text{cup } (2, \text{ } \backslash \text{infty })$
  • Function of Union ($\text{ } \backslash \text{cup }$): Unites both sets, signifying that any value $x$ lying in either interval is included in the solution set.
• The Intersection Symbol ($\text{ } \backslash \text{cap }$):
  • Represents the overlapping portion of two sets, corresponding to the logical connective "AND".
  • Bypassing in Standard Notation: The intersection symbol $\text{ } \backslash \text{cap }$ is rarely written explicitly in introductory algebra because overlapping "AND" inequalities condense directly into single intervals (such as $(-7, 3]$).
  • Intersection notation is explicitly written and utilized in specialized mathematics and teacher training laboratory courses.