Section sz.ineq

Polynomial & Rational Inequalities

Inequality problems appear on the final in both polynomial and rational forms. The sign-chart technique also comes back when you need to find domains of logarithmic functions.

⚡ Quick Summary

With a regular equation you're looking for exact values, but with an inequality you're looking for entire *regions* where something is positive or negative. Instead of asking "where does this equal zero?" you're asking "where is this above (or below) zero?" — and the answer is a range of values, not just a single point.

Sign Chart Method

\begin{aligned} &1.\;\text{Move everything to one side} \\ &2.\;\text{Factor completely} \\ &3.\;\text{Find critical points} \\ &4.\;\text{Test intervals} \end{aligned}

Critical points = zeros of numerator + zeros of denominator

Polynomial Inequality Setup

p(x) > 0 \;\text{ or }\; p(x) < 0 \;\text{ or }\; p(x) \geq 0 \;\text{ or }\; p(x) \leq 0

Always move everything to one side first so you're comparing to 0

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The key tool is the sign chart. Here's the idea: a polynomial or rational expression can only switch from positive to negative (or vice versa) at its critical points — the zeros and the undefined spots. Between those points, the sign stays the same. So you just need to find the critical points, pick one test value in each gap, and check whether the expression is positive or negative there.

The method follows a simple recipe: factor the expression completely, find all the zeros and undefined points, test one value in each interval, and write your answer in interval notation. Use brackets when the endpoint is included (the expression equals zero there and the inequality allows it), but always use parentheses at undefined points — you can never include a value where the expression doesn't exist.

Watch out for even-multiplicity roots like $(x - 2)^2$ — the expression touches zero there but doesn't actually change sign, just like a ball bouncing off the ground and going right back up. Also, never try to "multiply both sides" of a rational inequality by the denominator — you don't know if it's positive or negative, so you don't know which way the inequality flips. Stick with the sign chart.

Key Formulas

Sign Chart Method

\begin{aligned} &1.\;\text{Move everything to one side} \\ &2.\;\text{Factor completely} \\ &3.\;\text{Find critical points} \\ &4.\;\text{Test intervals} \end{aligned}

Critical points = zeros of numerator + zeros of denominator

Polynomial Inequality Setup

p(x) > 0 \;\text{ or }\; p(x) < 0 \;\text{ or }\; p(x) \geq 0 \;\text{ or }\; p(x) \leq 0

Always move everything to one side first so you're comparing to 0

Rational Inequality Setup

\frac{p(x)}{q(x)} \leq 0 \implies \text{critical points at } p(x)=0 \text{ and } q(x)=0

Never multiply both sides by the denominator — you don't know its sign

Interval Notation Reminders

[a, b] \text{ includes endpoints}, \quad (a, b) \text{ excludes endpoints}

Use ( ) at ±∞ and at values where the function is undefined

Sign Change Rule

\text{Sign changes at single roots; sign stays the same at double (even) roots}

A factor like (x - 2)² touches zero but doesn't change sign

Key Takeaways

✓Always use a sign chart: find critical points, test one value per interval, determine the sign
✓Critical points include both numerator zeros and denominator zeros — the sign can change at either
✓Use brackets $[\;]$ at zeros where the expression equals 0 (if the inequality allows equality); always use parentheses $(\;)$ at undefined points and at $\pm\infty$
✓Never multiply both sides of a rational inequality by the denominator — you don't know its sign

⚠️ Common Mistakes

✗Multiplying both sides of a rational inequality by the denominator — you don't know its sign, so the inequality might flip; always use a sign chart instead
✗Using a bracket at an undefined point — if $x = 3$ makes the denominator zero, ALWAYS use a parenthesis there, even with $\leq$ or $\geq$
✗Forgetting that even-multiplicity roots don't change the sign — $(x - 2)^2$ is always $\geq 0$ , so the expression doesn't flip sign at $x = 2$
✗Jumping straight to factoring without moving everything to one side first — you need the form $\text{expression} \leq 0$ (compared to zero) before factoring

Next: Section 4.1 Builds on: 3.7

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3.7 Rational Functions

4.1 Exponential Functions