Lesson 1-3: Polynomial & Rational Inequalities

Systematic Solving of Polynomial & Rational Inequalities

Learn a standardized, visual approach to solving inequalities of the form $f(x)\ge 0$ or $f(x)\le 0$ , including polynomial factorization, rational expression analysis, domain restrictions, sign charts, and careful solution verification.

Learning Objectives

Standardization

Transform inequality to $f(x)\gtreqless 0$ with zero on the right

Factorization

Factor polynomials into linear/quadratic factors to expose roots

Sign Chart Method

Use number lines and multiplicity to determine intervals of positivity/negativity

Domain & Verification

Exclude denominator zeros; verify boundary inclusion and domain validity

Polynomial Inequality Workflow

Move all terms to the left side: $f(x) \gtreqless 0$
Factor $f(x)$ into linear/quadratic factors. Determine roots and multiplicities.
Place all roots on a number line. Use the rule: odd multiplicity crosses (sign changes), even multiplicity bounces (sign preserved).
Select test points or use multiplicity logic to fill the sign chart.
Choose intervals where the sign meets the inequality ("positive" for > or ≥; "negative" for < or ≤).
Include boundary roots if the inequality is non-strict (≥ or ≤).

Worked Example: Polynomial Inequality

Solve $x^3-4x^2+x+6 \ge 0$ .

Factorization: test root x=2. $x^3-4x^2+x+6=(x-2)(x^2-2x-3)=(x-2)(x-3)(x+1)$
Roots: $x=-1, 2, 3$ (all single roots)
Sign chart: place points -1, 2, 3 on number line; alternate signs starting from rightmost interval.
Intervals meeting ≥ 0: $[-1,2] \cup [3, +\infty)$

Rational Inequality Workflow (Domain First!)

Standardize to $\tfrac{P(x)}{Q(x)} \gtreqless 0$ with denominator explicit; note $Q(x) \ne 0$ .
Factor numerator and denominator. Mark zeros of both on the number line. Denominator zeros are excluded from the solution set.
Use sign chart combining numerator sign and denominator sign: (+)/(+) and (-)/(-) give positive; (+)/(-) and (-)/(+) give negative.
Choose intervals satisfying the inequality; for non-strict inequalities, include numerator zeros (make expression 0) but never include denominator zeros.

Worked Example: Rational Inequality

Solve $\tfrac{x^2-2x-3}{x-2} \le 0$ .

Factor: numerator $(x-3)(x+1)$ , denominator $(x-2)$ .
Zeros: numerator at $x=-1, 3$ , denominator at $x=2$ (exclude x=2).
Sign chart and intervals: using sign analysis, solution is $\{-1\} \cup (2,3]$ .
Verification: check that $x=2$ is excluded; endpoints -1 and 3 satisfy equality (make LHS 0).

Real-World Applications of Inequalities

Feasibility Regions

In optimization, feasible sets are often defined by polynomial/rational inequalities (capacity, safety, or profit thresholds).

Use sign charts and boundary analysis to map admissible ranges for decision variables.

Physics & Engineering Safety

Stress/strain conditions, stability margins, and flow constraints can be captured by inequalities in design parameters.

Practice Problems

Problem 1: Polynomial Inequality

Solve $(x-4)(x+2)^2(x-1) \le 0$ and express the solution set using interval notation.

Problem 2: Rational Inequality

Solve $\tfrac{(x-1)(x+3)}{(x-2)(x+1)} \gt 0$ and specify excluded points.

Problem 3: Mixed Constraints

A system requires $x^2-5x+6 \ge 0$ and $\tfrac{x-1}{x-4} \le 0$ simultaneously. Find the admissible x-range.

← Back to Unit 1 Continue to Unit 2 →

Step-by-Step Technique Refinements

For quadratics, complete the square to visualize sign on intervals quickly.
For high-degree polynomials, synthetic division helps detect rational roots efficiently.
For irreducible quadratics, check discriminant to decide if factor sign is always positive or always negative.
In rational inequalities, always mark and exclude denominator zeros before making interval decisions.
Use a clean interval table recording the sign of each factor for clarity and fewer mistakes.

Additional Worked Examples

Example C

Solve $(x-2)^2(x+1) \gt 0$ .

Roots: x=2 (double), x=-1 (single). Even multiplicity at 2 means no sign change there.
Sign chart yields solution: $(-\infty,-1) \cup (2, +\infty)$ .

Example D

Solve $\tfrac{(x-3)(x+2)}{(x-1)^2} \le 0$ .

Zeros: numerator at x=3 and x=-2; denominator at x=1 (double, excluded).
Sign analysis shows solution: $[-2,1) \cup [3,3]$ (include -2 and 3; exclude 1; between 1 and 3 sign is positive for strictness).

Common Pitfalls & Sanity Checks

• Forgetting to flip inequality when multiplying by a negative factor (avoid algebraic manipulation of inequalities with unknown sign; prefer sign charts).
• Including denominator zeros in rational-inequality solutions (never allowed).
• Missing even-multiplicity bounces, leading to wrong interval signs.
• Ignoring domain constraints from the original problem context.

Practice Set II (with brief solutions)

4) Solve $(x^2-5x+6)(x-4) \ge 0$ .

Roots: 2, 3, 4 (all single). Sign chart ⇒ $(-\infty,2] \cup [3,4]$ .

5) Solve $\tfrac{x-2}{(x+1)(x-3)} \lt 0$ .

Critical points: -1, 2, 3. Exclude -1 and 3. Sign chart ⇒ $(-\infty,-1) \cup (2,3)$ .

6) Mixed: $x^3-4x \le 0$ and $\tfrac{x-1}{x+2} \gt 0$ simultaneously.

First: $x(x-2)(x+2) \le 0$ ⇒ intervals by sign chart; Second: exclude x=-2, positive when x>1 or x<-2. Intersect to obtain final range.

Challenge Problems (hints)

7) Solve $(x-1)^2(x^2+1) \ge 0$ .

Hint: $x^2+1 \gt 0$ for all x. Even multiplicity at x=1 means expression is nonnegative everywhere and zero at x=1.

8) Solve $\tfrac{x^2-4}{x^2+4x+4} \gt 0$ .

Hint: Factor and note denominator is a perfect square; analyze intervals excluding x=-2.

9) Solve $(x-3)^3(x+2)^2 \le 0$ .

Hint: Mix odd and even multiplicities to find sign; include even-root boundary if non-strict.

Recap

Standardize, factor, chart signs, apply domain, verify.
Odd multiplicity flips sign; even multiplicity preserves sign.
Rational inequalities: never include denominator zeros; include numerator zeros only for non-strict cases.

Theory Notes

If an irreducible quadratic $ax^2+bx+c$ has $\Delta=b^2-4ac < 0$ , then it never vanishes and its sign is the sign of $a$ .
For polynomials factored into real linear and irreducible quadratic factors, construct sign by multiplying signs of each factor on intervals between consecutive real roots.
For rational inequalities, the union of intervals must exclude all denominator roots; draw open circles at these points on the number line.

Guided Solutions

Example E

Solve $(x^2-1)(x-4) \lt 0$ .

Factor: $(x-1)(x+1)(x-4)$
Roots: -1, 1, 4 (all single). Sort: -1 < 1 < 4.
Sign chart across intervals: $(-\infty,-1),(-1,1),(1,4),(4,+\infty)$ .
Pick test points or alternate signs; select intervals where product < 0.
Solution: $(-\infty,-1) \cup (1,4)$

Example F

Solve $\tfrac{x^2-5x+6}{x^2-4} \ge 0$ .

Factor: numerator $(x-2)(x-3)$ , denominator $(x-2)(x+2)$ .
Simplify: cancels (x-2) but record a hole at x=2. Function becomes $\tfrac{x-3}{x+2}$ with domain x ≠ -2,2.
Zeros/undefined: zero at x=3 (include for ≥), vertical asymptote at x=-2, hole at x=2.
Sign chart on $(-\infty,-2),(-2,2),(2,3),(3,+\infty)$ for $\tfrac{x-3}{x+2}$ .
Solution: include intervals where expression ≥ 0 and x ≠ -2,2: $(-\infty,-2) \cup (2,3]$ .

Error Analysis

Algebraic manipulation of inequalities by multiplying both sides by expressions of unknown sign causes direction errors; prefer sign charts.
Endpoints in non-strict inequalities: include zeros of numerators but exclude denominator zeros and holes.
Order of critical points must be strictly increasing on the number line to avoid interval overlap mistakes.

Mixed Problems

10) Solve $(x-1)(x^2+1) \le (x+2)^2$ .

Bring to one side and factor where possible; note $x^2+1 \gt 0$ ; proceed with sign chart.

11) Solve $\tfrac{x-4}{x^2-1} \ge \tfrac{1}{x-1}$ .

Combine into a single rational expression with common denominator; mark x ≠ ±1; analyze signs.

12) Find x such that $|x-3| \le \tfrac{x^2-9}{x-3}$ .

Consider cases for x > 3, x < 3, and record x ≠ 3; simplify the rational piece to x+3 where defined.

Glossary

Sign Chart: Table or number line marking factor signs across intervals separated by critical points.
Multiplicity: Number of times a root repeats; even multiplicity preserves sign; odd multiplicity changes sign.
Hole: Removable discontinuity from cancelled common factors; not included in solution sets.
Vertical Asymptote: Non-removable infinite discontinuity at non-cancelled denominator roots.

Extended Applications

Safety Margins

Suppose allowable stress S satisfies $S(x)=S_0-\alpha x^2$ and must remain ≥ S_min. Solve $S_0-\alpha x^2 \ge S_{min}$ for design parameter x.

Throughput Constraints

If throughput $T(x)=\tfrac{x}{1+\beta x}$ must be ≤ T_max, solve the rational inequality for feasible x and exclude poles.

Step Checks

Did you bring everything to one side and simplify?
Did you factor completely and list all critical points in increasing order?
Did you exclude all denominator zeros from the solution?
Did you decide boundary inclusion correctly (≥/≤ include numerator zeros)?
Did you test a point in each interval if unsure?

Mini Quiz

Solve $(x-5)(x+1)^2 \ge 0$ .
Solve $\tfrac{x-2}{x^2-9} \le 0$ (state exclusions).
Solve $(x^2-4x+4)(x-1) \lt 0$ .

Answers

$(-\infty,-1] \cup [5,+\infty)$ (even multiplicity at -1, include boundaries).
$(-\infty,-3) \cup [2,3)$ with exclusions x=±3 (denominator zeros).
$(1, +\infty)$ since $(x-2)^2 \ge 0$ and need total < 0, so only where (x-1)<0 is false; check intervals to confirm.

Summary Checklist

Standardize → Factor → Chart → Decide → Verify.
Multiplicity matters for sign changes.
Rational inequalities require domain exclusions.

Note: When unsure, plot a few sample points to validate your sign chart decisions.