7: Solving Inequalities
- Page ID
- 173491
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 7.1: Solving Linear Inequalities
- This page explains linear inequalities, including definitions, properties, and solutions. It defines linear inequalities as expressions compared with relation symbols and discusses solutions as real numbers that make the inequality true. It covers the Addition and Multiplication Properties of Inequalities, manipulation techniques, graphical representations on a number line, and interval notation for solution sets, supplemented by step-by-step examples for better understanding.
- 7.2: Solving Compound Inequalities
- This page explains compound inequalities formed by "and" (conjunction) and "or" (disjunction), detailing their solution set intersections and unions. It covers three-part inequalities, solutions with correct notation, and theorems for manipulating inequalities through various operations. Real-world examples reinforce these concepts, highlighting scenarios such as disjoint and universal solutions.
- 7.3: Solving Absolute Value Inequalities
- This page explains absolute value inequalities, defining absolute value as distance from zero. It outlines how to convert these inequalities into compound "and" or "or" statements based on their structure. Theorems for each type are discussed, with examples illustrating applications, including solving inequalities and special cases related to negative numbers. The importance of isolating the absolute value before application of these rules is emphasized.
- 7.4: Solving Quadratic Inequalities
- This page summarizes quadratic inequalities, detailing definitions, key concepts, and solving methods. It explains expressions like \(ax^2+bx+c<0\) and defines terms such as boundary points and test intervals. Theorems clarify that quadratic expressions maintain a constant sign in intervals and how the discriminant affects the number of real zeros.
- 7.5: Solving Polynomial Inequalities
- This page covers polynomial inequalities, defining key terms like polynomial inequality, critical number, and zero multiplicity. It explains the Constant Sign Between Critical Numbers theorem, the Sign Behavior at a Zero theorem, and introduces the Sign-Chart Method for solving these inequalities.
- 7.6: Solving Rational Inequalities
- This page covers rational inequalities, detailing their definition, key concepts like boundary points, and the behavior of signs between these points. It explains solving techniques including expression rewriting, boundary point identification, interval testing, and selecting solutions based on signs.
- 7.7: Solving Exponential Inequalities
- This page discusses exponential inequalities, defining them with variables in exponents and explaining the monotonic behavior of exponential functions based on the base. It presents the order property for comparing exponential terms and offers strategies for solving inequalities, such as using common bases, natural logarithms, and interval testing with critical values. The page provides examples to illustrate these concepts.
- 7.8: Solving Logarithmic Inequalities
- This page discusses logarithmic inequalities, defining key concepts and outlining crucial theorems regarding positive arguments and comparisons of logarithms. It highlights the importance of the domain in logarithmic equations and presents two main theorems about logarithms in relation to constants and each other based on their base values. Additionally, it includes warnings about domain intersections and offers practical examples for solving logarithmic inequalities effectively.