Solving Polynomial and Rational Inequalities

The Logic of Solving Inequalities

Think about a function, $f(x)$. If we want to solve an inequality like $f(x)>0$, we're really asking, "For which x-values is the graph of this function above the x-axis?"

A continuous function can only change its sign (from positive to negative, or vice versa) by passing through zero. For rational functions, the sign can also flip on either side of a vertical asymptote (where the function is undefined). Therefore, the roots and vertical asymptotes are the critical points that divide the number line into regions where the function's sign is consistent.

Our entire method today rests on this simple idea: find the critical points, test the regions between them, and identify which regions satisfy the inequality.

1. Solving Polynomial Inequalities

Let's start with polynomials. A polynomial inequality might look like $x^2+3x>10$ or $x^3-x\le 3x^2-3$.

The process for solving them is systematic:

1. Set to Zero: Move all terms to one side of the inequality to get a polynomial $P(x)$ on one side and 0 on the other. For example, $P(x)>0$.
2. Find Critical Points: Find all the real roots of the polynomial $P(x)=0$. You might need to factorise (factor), use the quadratic formula, or even apply the Rational Root Theorem and synthetic division for higher-degree polynomials.
3. Create a Number Line: Draw a number line and mark your critical points on it.
   • Use closed circles (●) if the inequality is $\ge$ or $\le$ (inclusive of the roots).
   • Use open circles (○) if the inequality is $>$ or $<$ (exclusive of the roots).
4. Perform Sign Analysis: The critical points divide the line into intervals. Pick a convenient test number from within each interval and substitute it into the polynomial $P(x)$. Determine if the result is positive (+) or negative (-). This sign applies to the entire interval.
5. State the Solution: Identify the intervals that satisfy the inequality. Write your answer using interval notation or as an inequality.

Let's see this procedure in action.

As you saw, the method is a very direct application of the steps above.

Handling More Complex Polynomials

The same method works no matter how complex the polynomial is. The main challenge often lies in Step 2: finding the roots. This might require techniques we've learned in previous lessons.

This next video segment demonstrates solving a cubic inequality that first requires factoring by grouping, and then another that requires the Rational Root Theorem.

An interesting point was raised at the end of that video regarding multiplicity. If a factor $(x-c)$ has an odd power (e.g., $(x-c{)}^1,(x-c{)}^3$), the sign of the polynomial will flip as you cross the root $c$. If it has an even power (e.g., $(x-c{)}^2,(x-c{)}^4$), the polynomial touches the x-axis at $c$ but doesn't cross it, so the sign will be the same on both sides of the root. This can sometimes save you from testing every single interval.

2. Solving Rational Inequalities

Now let's turn to rational inequalities, such as $\frac{3-x}{x-1}<0$. The process is very similar, but with one crucial difference in finding the critical points.

A Critical Warning: Do Not Cross-Multiply!

It is tempting to solve $\frac{A(x)}{B(x)}<C$ by multiplying both sides by $B(x)$. You must resist this urge!

Why? Because we don't know if $B(x)$ is positive or negative. If it's negative, we would need to flip the inequality sign. Since its sign depends on $x$, we cannot multiply by it directly without potentially corrupting the solution.

The safe and correct method, as the resource points out, is to use critical values—which is exactly what we've been doing with sign analysis.

The Method for Rational Inequalities

1. Set to Zero: Move all terms to one side to get a single rational expression on one side and 0 on the other. You may need to find a common denominator to combine fractions.
2. Find Critical Points: Find all real values of $x$ where:
   • The numerator is zero (these are the roots of the function).
   • The denominator is zero (these are the vertical asymptotes, where the function is undefined).
3. Create a Number Line: Mark all critical points.
   • Points from the denominator (asymptotes) are ALWAYS open circles (○) because the function is undefined there.
   • Points from the numerator (roots) are open or closed based on the inequality ($>,<$ vs. $\ge ,\le$).
4. Perform Sign Analysis: Test a value from each interval in the simplified rational expression.
5. State the Solution: Write down the intervals that satisfy the inequality.

Let's watch a quick video that solves a rational inequality, connecting the process back to the graph.

An Alternative: The Table of Signs

For expressions with many factors, testing the entire polynomial or rational function can be cumbersome. A more organised (organized) approach is the Table of Signs.

Instead of testing the whole function, you create a table where the rows are the individual factors and the columns are the intervals from your number line. You then determine the sign of each factor in each interval. The final sign for the interval is found by combining the signs of the factors (e.g., a negative times a negative is a positive).

This method is particularly useful because it breaks a complex problem down into simpler parts, reducing the chance of calculation errors.

Conclusion

Today we formalised (formalized) the technique of sign analysis to solve both polynomial and rational inequalities. While the two types of functions are different, the underlying strategy is nearly identical and serves as a powerful capstone to our study of their properties.

Key Takeaways:

• The core method for solving these inequalities is sign analysis.
• Critical points are the values where the expression can change sign.
  • For polynomials, these are the roots.
  • For rational functions, these are the roots (from the numerator) AND the points of discontinuity (from the denominator).
• Always bring all terms to one side to compare the expression to zero.
• Never cross-multiply by a variable expression in a rational inequality.
• Pay close attention to whether the inequality is strict ($>,<$) or inclusive ($\ge ,\le$) to determine whether to use open (○) or closed (●) circles. Remember that critical points from a denominator are always open circles.

Preview of the Next Lesson:
We have now concluded our deep dive into polynomial and rational functions. In the next module, we will shift gears to explore a completely new and incredibly important class of functions: exponential and logarithmic functions. Our first lesson will be to evaluate exponential functions and analyse (analyze) their graphs and properties. You will see how these functions model phenomena like population growth and compound interest, and how their behaviour (behavior) is fundamentally different from the functions we've studied so far.