Polynomial Inequalities (Practice Problems) - Pauls Online Math Notes

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  • Algebra
    • 1. Preliminaries
      • 1.1 Integer Exponents
      • 1.2 Rational Exponents
      • 1.3 Radicals
      • 1.4 Polynomials
      • 1.5 Factoring Polynomials
      • 1.6 Rational Expressions
      • 1.7 Complex Numbers
    • 2. Solving Equations and Inequalities
      • 2.1 Solutions and Solution Sets
      • 2.2 Linear Equations
      • 2.3 Applications of Linear Equations
      • 2.4 Equations With More Than One Variable
      • 2.5 Quadratic Equations - Part I
      • 2.6 Quadratic Equations - Part II
      • 2.7 Quadratic Equations : A Summary
      • 2.8 Applications of Quadratic Equations
      • 2.9 Equations Reducible to Quadratic in Form
      • 2.10 Equations with Radicals
      • 2.11 Linear Inequalities
      • 2.12 Polynomial Inequalities
      • 2.13 Rational Inequalities
      • 2.14 Absolute Value Equations
      • 2.15 Absolute Value Inequalities
    • 3. Graphing and Functions
      • 3.1 Graphing
      • 3.2 Lines
      • 3.3 Circles
      • 3.4 The Definition of a Function
      • 3.5 Graphing Functions
      • 3.6 Combining Functions
      • 3.7 Inverse Functions
    • 4. Common Graphs
      • 4.1 Lines, Circles and Piecewise Functions
      • 4.2 Parabolas
      • 4.3 Ellipses
      • 4.4 Hyperbolas
      • 4.5 Miscellaneous Functions
      • 4.6 Transformations
      • 4.7 Symmetry
      • 4.8 Rational Functions
    • 5. Polynomial Functions
      • 5.1 Dividing Polynomials
      • 5.2 Zeroes/Roots of Polynomials
      • 5.3 Graphing Polynomials
      • 5.4 Finding Zeroes of Polynomials
      • 5.5 Partial Fractions
    • 6. Exponential and Logarithm Functions
      • 6.1 Exponential Functions
      • 6.2 Logarithm Functions
      • 6.3 Solving Exponential Equations
      • 6.4 Solving Logarithm Equations
      • 6.5 Applications
    • 7. Systems of Equations
      • 7.1 Linear Systems with Two Variables
      • 7.2 Linear Systems with Three Variables
      • 7.3 Augmented Matrices
      • 7.4 More on the Augmented Matrix
      • 7.5 Nonlinear Systems
  • Calculus I
    • 1. Review
      • 1.1 Functions
      • 1.2 Inverse Functions
      • 1.3 Trig Functions
      • 1.4 Solving Trig Equations
      • 1.5 Trig Equations with Calculators, Part I
      • 1.6 Trig Equations with Calculators, Part II
      • 1.7 Exponential Functions
      • 1.8 Logarithm Functions
      • 1.9 Exponential and Logarithm Equations
      • 1.10 Common Graphs
    • 2. Limits
      • 2.1 Tangent Lines and Rates of Change
      • 2.2 The Limit
      • 2.3 One-Sided Limits
      • 2.4 Limit Properties
      • 2.5 Computing Limits
      • 2.6 Infinite Limits
      • 2.7 Limits At Infinity, Part I
      • 2.8 Limits At Infinity, Part II
      • 2.9 Continuity
      • 2.10 The Definition of the Limit
    • 3. Derivatives
      • 3.1 The Definition of the Derivative
      • 3.2 Interpretation of the Derivative
      • 3.3 Differentiation Formulas
      • 3.4 Product and Quotient Rule
      • 3.5 Derivatives of Trig Functions
      • 3.6 Derivatives of Exponential and Logarithm Functions
      • 3.7 Derivatives of Inverse Trig Functions
      • 3.8 Derivatives of Hyperbolic Functions
      • 3.9 Chain Rule
      • 3.10 Implicit Differentiation
      • 3.11 Related Rates
      • 3.12 Higher Order Derivatives
      • 3.13 Logarithmic Differentiation
    • 4. Applications of Derivatives
      • 4.1 Rates of Change
      • 4.2 Critical Points
      • 4.3 Minimum and Maximum Values
      • 4.4 Finding Absolute Extrema
      • 4.5 The Shape of a Graph, Part I
      • 4.6 The Shape of a Graph, Part II
      • 4.7 The Mean Value Theorem
      • 4.8 Optimization
      • 4.9 More Optimization Problems
      • 4.10 L'Hospital's Rule and Indeterminate Forms
      • 4.11 Linear Approximations
      • 4.12 Differentials
      • 4.13 Newton's Method
      • 4.14 Business Applications
    • 5. Integrals
      • 5.1 Indefinite Integrals
      • 5.2 Computing Indefinite Integrals
      • 5.3 Substitution Rule for Indefinite Integrals
      • 5.4 More Substitution Rule
      • 5.5 Area Problem
      • 5.6 Definition of the Definite Integral
      • 5.7 Computing Definite Integrals
      • 5.8 Substitution Rule for Definite Integrals
    • 6. Applications of Integrals
      • 6.1 Average Function Value
      • 6.2 Area Between Curves
      • 6.3 Volumes of Solids of Revolution / Method of Rings
      • 6.4 Volumes of Solids of Revolution/Method of Cylinders
      • 6.5 More Volume Problems
      • 6.6 Work
    • Appendix A. Extras
      • A.1 Proof of Various Limit Properties
      • A.2 Proof of Various Derivative Properties
      • A.3 Proof of Trig Limits
      • A.4 Proofs of Derivative Applications Facts
      • A.5 Proof of Various Integral Properties
      • A.6 Area and Volume Formulas
      • A.7 Types of Infinity
      • A.8 Summation Notation
      • A.9 Constant of Integration
  • Calculus II
    • 7. Integration Techniques
      • 7.1 Integration by Parts
      • 7.2 Integrals Involving Trig Functions
      • 7.3 Trig Substitutions
      • 7.4 Partial Fractions
      • 7.5 Integrals Involving Roots
      • 7.6 Integrals Involving Quadratics
      • 7.7 Integration Strategy
      • 7.8 Improper Integrals
      • 7.9 Comparison Test for Improper Integrals
      • 7.10 Approximating Definite Integrals
    • 8. Applications of Integrals
      • 8.1 Arc Length
      • 8.2 Surface Area
      • 8.3 Center of Mass
      • 8.4 Hydrostatic Pressure
      • 8.5 Probability
    • 9. Parametric Equations and Polar Coordinates
      • 9.1 Parametric Equations and Curves
      • 9.2 Tangents with Parametric Equations
      • 9.3 Area with Parametric Equations
      • 9.4 Arc Length with Parametric Equations
      • 9.5 Surface Area with Parametric Equations
      • 9.6 Polar Coordinates
      • 9.7 Tangents with Polar Coordinates
      • 9.8 Area with Polar Coordinates
      • 9.9 Arc Length with Polar Coordinates
      • 9.10 Surface Area with Polar Coordinates
      • 9.11 Arc Length and Surface Area Revisited
    • 10. Series & Sequences
      • 10.1 Sequences
      • 10.2 More on Sequences
      • 10.3 Series - The Basics
      • 10.4 Convergence/Divergence of Series
      • 10.5 Special Series
      • 10.6 Integral Test
      • 10.7 Comparison Test/Limit Comparison Test
      • 10.8 Alternating Series Test
      • 10.9 Absolute Convergence
      • 10.10 Ratio Test
      • 10.11 Root Test
      • 10.12 Strategy for Series
      • 10.13 Estimating the Value of a Series
      • 10.14 Power Series
      • 10.15 Power Series and Functions
      • 10.16 Taylor Series
      • 10.17 Applications of Series
      • 10.18 Binomial Series
    • 11. Vectors
      • 11.1 Vectors - The Basics
      • 11.2 Vector Arithmetic
      • 11.3 Dot Product
      • 11.4 Cross Product
    • 12. 3-Dimensional Space
      • 12.1 The 3-D Coordinate System
      • 12.2 Equations of Lines
      • 12.3 Equations of Planes
      • 12.4 Quadric Surfaces
      • 12.5 Functions of Several Variables
      • 12.6 Vector Functions
      • 12.7 Calculus with Vector Functions
      • 12.8 Tangent, Normal and Binormal Vectors
      • 12.9 Arc Length with Vector Functions
      • 12.10 Curvature
      • 12.11 Velocity and Acceleration
      • 12.12 Cylindrical Coordinates
      • 12.13 Spherical Coordinates
  • Calculus III
    • 12. 3-Dimensional Space
      • 12.1 The 3-D Coordinate System
      • 12.2 Equations of Lines
      • 12.3 Equations of Planes
      • 12.4 Quadric Surfaces
      • 12.5 Functions of Several Variables
      • 12.6 Vector Functions
      • 12.7 Calculus with Vector Functions
      • 12.8 Tangent, Normal and Binormal Vectors
      • 12.9 Arc Length with Vector Functions
      • 12.10 Curvature
      • 12.11 Velocity and Acceleration
      • 12.12 Cylindrical Coordinates
      • 12.13 Spherical Coordinates
    • 13. Partial Derivatives
      • 13.1 Limits
      • 13.2 Partial Derivatives
      • 13.3 Interpretations of Partial Derivatives
      • 13.4 Higher Order Partial Derivatives
      • 13.5 Differentials
      • 13.6 Chain Rule
      • 13.7 Directional Derivatives
    • 14. Applications of Partial Derivatives
      • 14.1 Tangent Planes and Linear Approximations
      • 14.2 Gradient Vector, Tangent Planes and Normal Lines
      • 14.3 Relative Minimums and Maximums
      • 14.4 Absolute Minimums and Maximums
      • 14.5 Lagrange Multipliers
    • 15. Multiple Integrals
      • 15.1 Double Integrals
      • 15.2 Iterated Integrals
      • 15.3 Double Integrals over General Regions
      • 15.4 Double Integrals in Polar Coordinates
      • 15.5 Triple Integrals
      • 15.6 Triple Integrals in Cylindrical Coordinates
      • 15.7 Triple Integrals in Spherical Coordinates
      • 15.8 Change of Variables
      • 15.9 Surface Area
      • 15.10 Area and Volume Revisited
    • 16. Line Integrals
      • 16.1 Vector Fields
      • 16.2 Line Integrals - Part I
      • 16.3 Line Integrals - Part II
      • 16.4 Line Integrals of Vector Fields
      • 16.5 Fundamental Theorem for Line Integrals
      • 16.6 Conservative Vector Fields
      • 16.7 Green's Theorem
    • 17.Surface Integrals
      • 17.1 Curl and Divergence
      • 17.2 Parametric Surfaces
      • 17.3 Surface Integrals
      • 17.4 Surface Integrals of Vector Fields
      • 17.5 Stokes' Theorem
      • 17.6 Divergence Theorem
  • Differential Equations
    • 1. Basic Concepts
      • 1.1 Definitions
      • 1.2 Direction Fields
      • 1.3 Final Thoughts
    • 2. First Order DE's
      • 2.1 Linear Equations
      • 2.2 Separable Equations
      • 2.3 Exact Equations
      • 2.4 Bernoulli Differential Equations
      • 2.5 Substitutions
      • 2.6 Intervals of Validity
      • 2.7 Modeling with First Order DE's
      • 2.8 Equilibrium Solutions
      • 2.9 Euler's Method
    • 3. Second Order DE's
      • 3.1 Basic Concepts
      • 3.2 Real & Distinct Roots
      • 3.3 Complex Roots
      • 3.4 Repeated Roots
      • 3.5 Reduction of Order
      • 3.6 Fundamental Sets of Solutions
      • 3.7 More on the Wronskian
      • 3.8 Nonhomogeneous Differential Equations
      • 3.9 Undetermined Coefficients
      • 3.10 Variation of Parameters
      • 3.11 Mechanical Vibrations
    • 4. Laplace Transforms
      • 4.1 The Definition
      • 4.2 Laplace Transforms
      • 4.3 Inverse Laplace Transforms
      • 4.4 Step Functions
      • 4.5 Solving IVP's with Laplace Transforms
      • 4.6 Nonconstant Coefficient IVP's
      • 4.7 IVP's With Step Functions
      • 4.8 Dirac Delta Function
      • 4.9 Convolution Integrals
      • 4.10 Table Of Laplace Transforms
    • 5. Systems of DE's
      • 5.1 Review : Systems of Equations
      • 5.2 Review : Matrices & Vectors
      • 5.3 Review : Eigenvalues & Eigenvectors
      • 5.4 Systems of Differential Equations
      • 5.5 Solutions to Systems
      • 5.6 Phase Plane
      • 5.7 Real Eigenvalues
      • 5.8 Complex Eigenvalues
      • 5.9 Repeated Eigenvalues
      • 5.10 Nonhomogeneous Systems
      • 5.11 Laplace Transforms
      • 5.12 Modeling
    • 6. Series Solutions to DE's
      • 6.1 Review : Power Series
      • 6.2 Review : Taylor Series
      • 6.3 Series Solutions
      • 6.4 Euler Equations
    • 7. Higher Order Differential Equations
      • 7.1 Basic Concepts for nth Order Linear Equations
      • 7.2 Linear Homogeneous Differential Equations
      • 7.3 Undetermined Coefficients
      • 7.4 Variation of Parameters
      • 7.5 Laplace Transforms
      • 7.6 Systems of Differential Equations
      • 7.7 Series Solutions
    • 8. Boundary Value Problems & Fourier Series
      • 8.1 Boundary Value Problems
      • 8.2 Eigenvalues and Eigenfunctions
      • 8.3 Periodic Functions & Orthogonal Functions
      • 8.4 Fourier Sine Series
      • 8.5 Fourier Cosine Series
      • 8.6 Fourier Series
      • 8.7 Convergence of Fourier Series
    • 9. Partial Differential Equations
      • 9.1 The Heat Equation
      • 9.2 The Wave Equation
      • 9.3 Terminology
      • 9.4 Separation of Variables
      • 9.5 Solving the Heat Equation
      • 9.6 Heat Equation with Non-Zero Temperature Boundaries
      • 9.7 Laplace's Equation
      • 9.8 Vibrating String
      • 9.9 Summary of Separation of Variables
  • Extras
  • Algebra & Trig Review
    • 1. Algebra
      • 1.1 Exponents
      • 1.2 Absolute Value
      • 1.3 Radicals
      • 1.4 Rationalizing
      • 1.5 Functions
      • 1.6 Multiplying Polynomials
      • 1.7 Factoring
      • 1.8 Simplifying Rational Expressions
      • 1.9 Graphing and Common Graphs
      • 1.10 Solving Equations, Part I
      • 1.11 Solving Equations, Part II
      • 1.12 Solving Systems of Equations
      • 1.13 Solving Inequalities
      • 1.14 Absolute Value Equations and Inequalities
    • 2. Trigonometry
      • 2.1 Trig Function Evaluation
      • 2.2 Graphs of Trig Functions
      • 2.3 Trig Formulas
      • 2.4 Solving Trig Equations
      • 2.5 Inverse Trig Functions
    • 3. Exponentials & Logarithms
      • 3.1 Basic Exponential Functions
      • 3.2 Basic Logarithm Functions
      • 3.3 Logarithm Properties
      • 3.4 Simplifying Logarithms
      • 3.5 Solving Exponential Equations
      • 3.6 Solving Logarithm Equations
  • Common Math Errors
    • 1. General Errors
    • 2. Algebra Errors
    • 3. Trig Errors
    • 4. Common Errors
    • 5. Calculus Errors
  • Complex Number Primer
    • 1. The Definition
    • 2. Arithmetic
    • 3. Conjugate and Modulus
    • 4. Polar and Exponential Forms
    • 5. Powers and Roots
  • How To Study Math
    • 1. General Tips
    • 2. Taking Notes
    • 3. Getting Help
    • 4. Doing Homework
    • 5. Problem Solving
    • 6. Studying For an Exam
    • 7. Taking an Exam
    • 8. Learn From Your Errors
  • Cheat Sheets & Tables
  • Misc Links
  • Contact Me
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Section 2.12 : Polynomial Inequalities

Solve each of the following inequalities.

  1. \({u^2} + 4u \ge 21\) Solution
  2. \({x^2} + 8x + 12 < 0\) Solution
  3. \(4{t^2} \le 15 - 17t\) Solution
  4. \({z^2} + 34 > 12z\) Solution
  5. \({y^2} - 2y + 1 \le 0\) Solution
  6. \({t^4} + {t^3} - 12{t^2} < 0\) Solution
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