Rational Inequalities

Definition

A rational inequality is an expression that compares a rational function to zero (e.g., $R(x) \le 0$). Solving it involves finding the intervals of $x$ for which the inequality holds true.

• How to read: “The rational function R of x is less than or equal to zero.”
• Meaning: Find all $x$ where the rational expression is positive, negative, or zero using sign analysis on intervals.

Why It Matters

Rational functions often model systems with rates or ratios. If you miss a single “cut point” (like a zero in the denominator), you might think a system is stable when it’s actually about to “break” (become undefined). It is the math of boundary analysis with constraints.

Core Concepts

• Boundary Points (Cut Points): The values of $x$ where the function can change sign. For rational functions, these are the real zeros of the numerator AND the real zeros of the denominator.
• Interval Testing: The real number line is divided into intervals by the boundary points. The function’s sign remains constant within each interval.
• Strict vs. Non-Strict:
  • $> or <$: Boundary points are generally excluded (open circles/parentheses).
  • $\ge or \le$: Zeros of the numerator are included (closed circles/brackets). Zeros of the denominator are NEVER included because they make the expression undefined.
  • How to read: “The strict inequalities greater than or less than; and the non-strict inequalities greater than or equal to, or less than or equal to.”
  • Meaning / when to use: Include numerator zeros only for $\ge/\le$; denominator zeros are always excluded since they represent vertical asymptotes or holes.

Connected Concepts

• Polynomial Inequalities
• Interval Notation
• Rational Functions
• First Principles Thinking