Polynomial and Rational Inequalities: Methods and Examples

Section 5.5: Polynomial and Rational Inequalities

Introduction

Polynomial and rational inequalities are important topics in precalculus, as they extend the concept of solving equations to finding the set of values for which a function is greater than or less than zero. This section covers graphical and algebraic methods for solving such inequalities, including step-by-step procedures and illustrative examples.

Solving Polynomial Inequalities Graphically

Graphical methods involve analyzing the graph of a polynomial function to determine where the function is positive or negative.

• Key Point: The solution to a polynomial inequality corresponds to the intervals where the graph lies above or below the x-axis, depending on the inequality sign.

• Example: If asked to solve a polynomial inequality graphically, plot the function and identify the x-values where the function meets the required condition (e.g., $f(x) \geq 0$ means $f(x)$ is on or above the x-axis).

Solving Rational Inequalities Graphically

Rational inequalities involve functions that are ratios of polynomials. The graphical approach is similar, but special attention must be paid to points where the function is undefined (vertical asymptotes).

• Key Point: The solution set excludes values where the denominator is zero, as the function is undefined at those points.

• Example: To solve a rational inequality graphically, plot the function, note any vertical asymptotes, and identify intervals where the function satisfies the inequality.

Steps for Solving Polynomial and Rational Inequalities Algebraically

Algebraic methods provide a systematic approach to solving inequalities without graphing.

1. Step 1: Write the inequality so that a polynomial or rational function is on the left side and zero is on the right side, in one of the following forms:

   • $f(x) \gt 0$

   • $f(x) \lt 0$

   • $f(x) \geq 0$

   • $f(x) \leq 0$

   For rational functions: Ensure the left side is written as a single rational expression, then find the domain of $f$.

2. Step 2: Determine the real numbers at which $f(x) = 0$ (the zeros of the function), and for rational functions, also find the real numbers where $f$ is undefined (denominator equals zero).

3. Step 3: Use the numbers found in Step 2 to divide the real number line into intervals.

4. Step 4: Select a number in each interval and evaluate $f$ at that number.

   • If $f$ is positive at the test point, then the inequality holds for all $x$ in that interval (for $f(x) \gt 0$ or $f(x) \geq 0$).

   • If $f$ is negative at the test point, then the inequality holds for all $x$ in that interval (for $f(x) \lt 0$ or $f(x) \leq 0$).

Examples of Solving Polynomial and Rational Inequalities Algebraically

Below are examples illustrating the algebraic solution process:

• Example 4: Solve $x^3 - x^2 - 90x \geq 0$ algebraically.

  • Factor the polynomial: $x^3 - x^2 - 90x = x(x^2 - x - 90)$

  • Factor further: $x^2 - x - 90 = (x - 10)(x + 9)$

  • So, $x^3 - x^2 - 90x = x(x - 10)(x + 9)$

  • Set each factor to zero: $x = 0$, $x - 10 = 0 \Rightarrow x = 10$, $x + 9 = 0 \Rightarrow x = -9$

  • Divide the real line at $x = -9$, $x = 0$, $x = 10$ into intervals: $(-\infty, -9)$, $(-9, 0)$, $(0, 10)$, $(10, \infty)$

  • Test a value in each interval to determine where the product is positive or zero.

  • Solution: The set of $x$ values where $x^3 - x^2 - 90x \geq 0$.

• Example 5: Solve $x - 4x + 2 \geq 0$ algebraically.

  • Assuming the intended expression is a rational function, e.g., $\frac{x - 4}{x + 2} \geq 0$ (Additional info: inferred from context).

  • Find zeros: $x - 4 = 0 \Rightarrow x = 4$

  • Find undefined points: $x + 2 = 0 \Rightarrow x = -2$

  • Divide the real line at $x = -2$, $x = 4$ into intervals: $(-\infty, -2)$, $(-2, 4)$, $(4, \infty)$

  • Test a value in each interval to determine where the expression is positive or zero.

  • Solution: The set of $x$ values where $\frac{x - 4}{x + 2} \geq 0$.

• Example 6: Solve $(x - 2)(x + 9)x \leq 0$ algebraically.

  • Set each factor to zero: $x - 2 = 0 \Rightarrow x = 2$, $x + 9 = 0 \Rightarrow x = -9$, $x = 0$

  • Divide the real line at $x = -9$, $x = 0$, $x = 2$ into intervals: $(-\infty, -9)$, $(-9, 0)$, $(0, 2)$, $(2, \infty)$

  • Test a value in each interval to determine where the product is negative or zero.

  • Solution: The set of $x$ values where $(x - 2)(x + 9)x \leq 0$.

Summary Table: Steps for Solving Polynomial and Rational Inequalities

Step	Description
1	Rewrite the inequality with zero on one side and the function on the other.
2	Find zeros of the function and, for rational functions, points where the function is undefined.
3	Divide the real number line into intervals using the values found in Step 2.
4	Test a value in each interval to determine where the inequality holds.

Additional info: Some expressions in the examples were inferred to be rational functions based on context and standard precalculus practice.