Solving Polynomial and Rational Inequalities by Sign Analysis

1. The Challenge of Non-Linear Inequalities

When you solve a linear inequality like $2x-1>5$, you can simply isolate $x$ just as you would with an equation, with the only extra rule being to flip the inequality sign if you multiply or divide by a negative number.

However, this approach fails for inequalities like $x^2-3x>10$. Why? If we rearrange it to $(x-5)(x+2)>0$, we might be tempted to divide by, say, $(x-5)$. But we don't know if $(x-5)$ is positive or negative, as its sign depends on the value of $x$ we're trying to find! This is a circular problem.

To solve this properly, we need a more robust method that doesn't rely on isolating the variable. The key insight is that a function can only change its sign (from positive to negative or vice versa) at points where it is either zero or undefined. These special points are our critical values.

2. Solving Polynomial Inequalities

A polynomial inequality involves a polynomial expression with an inequality sign, for example, $x^3-x\le 3x^2-3$. The method is straightforward and systematic.

The Four-Step Method:

1. Get Zero on One Side: Rearrange the inequality so that one side is zero. For example, $x^2-3x-10>0$. Let's call the polynomial expression $f(x)$.
2. Find Critical Values: Find the roots of the polynomial by solving $f(x)=0$. These are the only values where the polynomial can change its sign.
3. Create a Sign Chart:
   • Draw a number line and mark the critical values you found. These values divide the line into several intervals.
   • Pick a convenient test value from within each interval.
   • Substitute each test value into the factored form of the polynomial $f(x)$ and determine if the result is positive (+) or negative (-). You only need the sign, not the exact value.
4. Determine the Solution: Based on the sign chart and the original inequality (e.g., > 0 means you want positive intervals), identify the intervals that satisfy the inequality. Write your final answer using interval notation, being careful with parentheses () for strict inequalities (<, >) and square brackets [] for inclusive inequalities (<=, >=).

The following resource provides an excellent walkthrough of this method with clear, step-by-step examples.

To see this process in action, the following video offers several examples, starting with a simple quadratic and moving to more complex cubics.

A Note on Repeated Roots (Multiplicity)

A common mistake is to assume that the signs on the number line will always alternate (+, -, +, ...). This is not always true! This assumption fails when a polynomial has a root with an even multiplicity, for example, the root $x=3$ in $f(x)=(x-3{)}^2(x+1)$.

At such a root, the graph of the function touches the x-axis but doesn't cross it, so the sign of $f(x)$ is the same on both sides of the root.

The takeaway: You must test a value from every interval. Do not assume alternation.

The following resource provides good examples of this situation.

3. Solving Rational Inequalities

The method for solving rational inequalities like $\frac{x+3}{(x+2)(x-2)}\le 0$ is very similar to the polynomial method, but with one crucial addition.

Besides the roots (where the numerator is zero), the sign of a rational function can also change at its vertical asymptotes (where the denominator is zero). Therefore, our set of critical values must include values from both the numerator and the denominator.

The Modified Method:

1. Get Zero and a Single Fraction: Rearrange the inequality to have zero on one side, and if necessary, combine terms to form a single algebraic fraction. Crucial warning: Never multiply both sides by an expression involving the variable (e.g., by $x-4$), because you don't know if its sign is positive or negative.
2. Find Critical Values: Identify all critical values by setting both the numerator and the denominator equal to zero and solving.
3. Create a Sign Chart: The process is the same. Mark all critical values on a number line and test the sign of the function in each interval.
4. Determine the Solution: Identify the solution intervals. Pay close attention to the endpoints:
   • Critical values from the numerator (roots) can be included if the inequality is inclusive ($\le$ or $\ge$).
   • Critical values from the denominator (restrictions/asymptotes) are never included in the solution set because the function is undefined there. They always get an open circle or parenthesis.

The resource we looked at earlier also covers this topic excellently.

This video also provides a concise visual walkthrough of the process.

Conclusion

Today, we've built a powerful and reliable method for solving non-linear inequalities. While it may seem like a multi-step process, its logic is consistent and it provides a clear framework for showing your reasoning, which is essential for earning method marks on your IB exams.

Key Takeaways:

• To solve polynomial or rational inequalities, always start by getting zero on one side and, for rational inequalities, combining everything into a single fraction.
• Identify all critical values. For polynomials, these are the roots. For rational functions, these are the roots (from the numerator) and the restrictions (from the denominator).
• Use a sign chart by plotting critical values on a number line and testing a point from each resulting interval. Never assume the signs will alternate.
• Write your final answer in interval notation. Remember that critical values from a denominator are never included in the solution set.

In our next lesson, we will learn about decomposing rational expressions into partial fractions. This is another key algebraic skill from your AA HL syllabus, and it serves as a direct prerequisite for an important integration technique you'll encounter later in your study of calculus.