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Transcript of DESIGUALDADES 1A
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Review IndexDesigualdades de valor absolutoUna desigualdad de valor absoluto es una desigualdad que tiene un signo de valor absoluto con una variable dentro.Desigualdades de valor absoluto ( -4 Y x < 4. El conjunto solucin es .
Cuando se resuelven desigualdes de valor absoluto, hay dos casos a considerar.Caso 1: La expresin dentro de los smbolos de valor absoluto es positiva.Caso 2: La expresin dentro de los smbolos de valor absoluto es negativa.La solucin es la interseccin de las soluciones de estos dos casos.En otras palabras, para cualesquiera numros reales a y b , si | a | < b, entonces a < b Y a > - b .
Ejemplo 1:Resuelva y grafique.|x 7| < 3
Para resolver este tipo de desigualdad, necesitamos descomponerla en una desigualdad compuesta.x 7 < 3 Y x 7 > 33 < x 7 < 3
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Sume 7 en cada expresin.-3 + 7 < x - 7 + 7 < 3 + 74 < x ):La desigualdad | x | > 4 significa que la distancia entre x y 0 es mayor que 4.
As, x < -4 O x > 4. El conjunto solucin es .Cuando se resuelven desigualdes de valor absoluto, hay dos casos a considerar.Caso 1: La expresin dentro de los smbolos de valor absoluto es positiva.Caso 2: La expresin dentro de los smbolos de valor absoluto es negativa.En otras palabras, para cualesquiera numros reales a y b , si | a | > b, entonces a > b O a < - b .
Ejemplo 2:Resuelva y grafique.
Separe en dos desigualdades.
Reste 2 de cada lado en cada desigualdad.
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La grfica se vera as:
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Review Topics
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Review Index
Review TopicsThis guide provides an alphabetical index of review topics.
A
l AA Similarityl AAS Postulatel Absolute Valuel Absolute Value of
a Complex Number
l Absolute Value Functions
l Absolute Value Inequalities
l Adding and Subtracting Complex Numbers
l Adding and Subtracting Decimals
l Adding and Subtracting Fractions
l Adding and Subtracting Matrices
l Adding and Subtracting Negatives
l Adding and Subtracting Rational Expressions with Like Denominators
l Adding and Subtracting Rational Expressions with
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Review Topics
Unlike Denominators
l Adding and Subtracting Vectors
l Adding and Subtracting Polynomials
l Addition Rule of Probability
l Additive Inversel Adjacent Anglesl Adjoint of a
Matrixl Alternate Exterior
Anglesl Alternate Exterior
Angles Theoreml Alternate Interior
Anglesl Alternate Interior
Angles Theoreml Altitude of a
Trianglel Amplitude and
Period of Sine and Cosine Functions
l Angle Addition Postulate
l Angle-Angle Similarity
l Angle Bisector Theorem
l Angle Bisectorl Angle of
Intersecting Chords Theorem
l Angle of Intersecting Secants Theorem
l Angle of Rotationl Angle Sum
Theoreml Angles of
Elevation and Depression
l Anglesl Arcsl Areal Area of a Circle
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Review Topics
l Area of a Trianglel Areas of Regular
Polygonsl Arithmetic
Sequencesl Arithmetic Seriesl ASA Postulatel Associative
Propertyl Associative
Property of Matrices
l Asymptotesl Augmented
Matricesl Axiom l Axis of
Symmetry of a Parabola
B
l Bar Graphsl Base (Exponential
Notation)l Base (Logarithms)l Base (Number
Systems)l Basic Operations l Basic
Trigonometric Identities
l Big and Small Numbers
l Binary Numbers l Binomial
Probability l Binomials l Binomial Seriesl Binomial Theoreml Bisectors in a
Triangle l Box and Whisker
Plots
C
l Cartesian Planel Center and
Variation of Datal Center of
Rotation l Central Angles
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Review Topics
l Centroidl Choosing
Appropriate Units of Measure
l Chord l Circlel Circle Graphs l Circles Inscribed
in Squares l Circumcenter
Theorem l Circumference l Cofunction and
Odd-Even Identities
l Collinear Pointsl Combinations l Combined
Variation l Combining Like
Terms with Exponents
l Combining Like Terms
l Common Difference
l Common Ratiol Commutative
Propertyl Comparing
Decimalsl Comparing
Fractionsl Comparing
Functionsl Compatible
Matricesl Complement of
an Eventl Complementary
Anglesl Completing the
Squarel Complex Numbersl Components of a
Vectorl Composite
Numbersl Composition of
Functionsl Compound
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Review Topics
Inequalitiesl Compound
Interestl Conditional
Probabilityl Conditional
Statementsl Conel Congruent Anglesl Congruent
Segmentsl Congruent
Tangents and Circumscribed Polygons
l Congruent Triangles on the Coordinate Plane
l Congruent Triangles
l Conic Sections and Standard Forms of Equations
l Conjugatesl Conjunctionl Consecutive
Interior Angles Theorem
l Consistent and Dependent Systems
l Constant Functionl Constant of
Variationl Converse, Inverse
and Contrapositivel Converse of
Isosceles Triangle Theorem
l Convert between Systems
l Convert Units of Area and Volume
l Converting Fractions to Decimals
l Converting Non Repeating Decimals to Fractions
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Review Topics
l Converting Repeating Decimals to Fractions
l Coordinate Proofsl Coplanarl Correlation and
Causal Relationl Corresponding
Anglesl Corresponding
Angles Postulatel Cosine Functionl Cosinel Coterminal Anglesl Counter Examplel Cramer's Rulel Cross Productsl Cross Sectionsl Cubic Functionsl Customary Unitsl Cylinder
D
l Decimals, Adding and Subtracting
l Decimals, Multiplying and Dividing
l Decimals, Terminating and Repeating
l Decimals, Rounding
l Decimals and Percentages
l Degree (of a Monomial)
l Degree (of a Polynomial)
l Degree (of a Vertex)
l Degree Measure of an Angle
l Degree to Radian Measure
l Denominatorl Dependent Eventsl Descartes Rule of
Signsl Describing the
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Review Topics
Graph of a Function
l Determinantsl Developing a
Probability Distribution from Empirical Data
l Diagonals of Parallelograms
l Diameter l Difference of
Squares l Different Basesl Dilationl Direct and
Inverse Variationl Direct Variation l Directrixl Discriminantsl Disjunctionl Distance between
Parallel Linesl Distance Formula
in 3Dl Distance Formulal Distributive
Property of Matrices
l Distributive Property
l Dividing by a Fraction
l Dividing Complex Numbers
l Dividing Fractionsl Dividing
Polynomialsl Dividing Rational
Expressionsl Divisibility Testsl Domains l Domain and
Rangel Double-Angle
and Half-Angle Identities
E
l e (the Number) l Eccentricityl Ellipse
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Review Topics
l End Behavior of a Function
l Endvertex (Graph Theory)
l Equation of a Circle
l Equivalent Fractions
l Evaluating Expressions
l Even and Odd Functions
l Expected Valuel Exponent Tables
and Patternsl Exponential
Decayl Exponential
Functionsl Exponential
Growthl Exponential
Regressionl Exponential to
Logarithmic Forml Exponentsl Exterior Angle
Inequalityl Exterior Angle
Theoreml Extraneous
Solutions
F
l Factor Theoreml Factorialsl Factoringl Factoring by
Groupingl Factoring
Monomialsl Factorsl Falling Objectsl Fibonacci
Numbersl Finding Arc
Lengthsl Finding the
Equation of a Line from Its Graph
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Review Topics
l Finding the Equation of a Parabola given Focus and Directrix
l Finite Differencesl Fitting Equations
to Datal Fitting Linear
Equations to Datal Focus of a
Parabolal Focusl Formulasl 45-45-90
Trianglesl Fraction
Operationsl Fractionsl Frequency
Distributionsl Functions
Describing Behavior
l Functionsl Fundamental
Counting Principlel Fundamental
Theorem of Algebra
G
l Gathering Like Terms
l Geometric Meanl Geometric
Probabilityl Geometric
Sequencesl Geometric Seriesl Golden Ratiol Golden Rectanglel Graphing
Exponential and Logarithmic Functions
l Graph Theoryl Graphing Cosine
Functionl Graphing
Inequalities in
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One Variablel Graphing Linear
Equations l Graphing Linear
Inequalities in Two Variables
l Graphing Polynomial Functions
l Graphing Quadratic Equations
l Graphing Quadratic Inequalities
l Graphing Rational Functions
l Graphing Sine Function
l Graphing Square Root Functions
l Graphing Tangent Function
l Greatest Common Factors (GCFs)
l Greatest Common Factor of Monomials
H
l Hexadecimal Numbers
l Histogramsl Horizontal Linesl Hyperbolasl Hypotenuse
I
l Identity Matrixl Improper
Fractionsl Independent/
Dependent Eventsl Independent
Eventsl Indirect Proofl Inductive
Reasoningl Inequalitiesl Infinite
Geometric Series
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Review Topics
l Infinityl Input-Output
Tablesl Inscribed Anglesl Inscribing a
Quadrilaterall Integersl Interpreting Datal Interquartilel Intersecting
Chords Theoreml Intersecting
Chordsl Intersecting
Secant-Tangent Theorem
l Intersecting Secants Theorem
l Interval Notationl Inverse Functionsl Inverse Matricesl Inverse Operationsl Inverse
Trigonometric Functions
l Inverse Variationl Irrational
Numbersl Irreducible
Polynomialsl Isosceles Triangle
Theorem
J
l Joint Variation
K
l Kite
L
l Latus Rectuml Law of Cosinesl Law of
Detachmentl Law of Sinesl Law of Syllogisml Leading
Coefficient Testl Least Common
Denominators
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Review Topics
(LCDs)l Least Common
Multiples (LCMs)l Line Graphsl Line of best fitl Line of best fit
(Eyeball Method)l Line Plots l Line Symmetryl Linear Equations,
Graphingl Linear Equations,
Solvingl Linear Equations,
Systems ofl Linear Functionl Linear Pair l Linear
Programming l Linesl Logarithmic
Functionsl Logarithmic to
Exponential Form l Logarithmsl Lowest Terms
M
l Magnitude and Direction of Vectors
l Major and Minor Axes of Conics
l Mapping Diagrams
l Matricesl Matrix
Dimensionsl Matrix
Multiplication l Matrix Row
Operations l Mean l Medianl Medians of a
Trianglel Metric Systeml Midpoint Formulal Midsegment of a
Trapezoidl Mixed
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Review Topics
Expressionsl Mixed Numbers l Model Monomialsl More on
Hyperbolasl Multiplication
Rule of Probability
l Multiplicative Inverse
l Multiplying a Fraction by a Fraction
l Multiplying a Fraction by an Integer
l Multiplying and Dividing with Decimals
l Multiplying and Dividing with Negatives
l Multiplying Complex Numbers
l Multiplying Monomials
l Multiplying Polynomials
l Multiplying Rational Expressions
l Multiplying Vector by a Matrix
l Mutually Exclusive Events
N
l Natural Numbersl Negationl Negative
Exponentsl Netsl Normal
Distribution of data
l nth Rootsl nth Term of a
Geometric Sequence
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Review Topics
l nth Term of an Arithmetic Sequence
l nth Term of a Sequence
l Number Line l Number Systemsl Numerator
O
l Odds (Probability)l One to One
Functionsl Operations on
Functionsl Operations on
Mixed Numbersl Operations on Setsl Operations with
Complex Numbers
l Operations with Decimals
l Operations with Fractions
l Operations with Negatives
l Oppositesl Ordered Pairl Outliersl Order of
Operations
P
l Parabolasl Parallel Lines and
Transversalsl Parallel Linesl Parallelograml Parametric
Equationsl Parent Graphsl Pascal's Trianglel Percentl Percentilel Percent Increase
and Decreasel Percents and
Decimals l Perfect Numbersl Perfect Squares
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Review Topics
l Perimeterl Perimeter, Area
and Volumel Periodic Functionsl Permutationsl Perpendicular
Bisectorl Perpendicular
Lines and Slopesl Perpendicular
Lines l Perpendicular
Transversal Theorem
l pi (Pi)l Piecewise-
Defined Functionl Planesl Platonic Solidsl Point-Slope Form
(of a Linear Equation)
l Polar Coordinates l Polar Form of a
Complex Number l Polygon Exterior
Angle Sum Theorem
l Polygon Interior Angles Sum Theorem
l Polygonsl Polyhedral Polynomial
Functionl Polynomial with
Complex Rootsl Power of a Power
Propertyl Power of a
Product Propertyl Power of a
Quotient Propertyl Powers of 10 l Powers of il Prime and
Composite Numbers
l Prime Factorization
l Prime Numbers
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Review Topics
l Principal Value of a Square Root
l Prism l Probability
Distributionl Probability
Modelsl Probabilityl Product of a Sum
and a Differencel Product of
Powers Propertyl Properties of
Addition l Properties of
Congruencel Properties of
Equalityl Properties of
Exponentsl Properties of
Inequalityl Properties of
Multiplication l Properties of
Square Roots l Proportionsl Pyramidl Pythagorean
Theorem l Pythagorean
Triples
Q
l Quadrantsl Quadratic
Equations l Quadratic
Equations, Solving by Completing the Square
l Quadratic Equations, Solving by Factoring
l Quadratic Equations, Solving by Square Roots
l Quadratic Formula
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Review Topics
l Quadratic Function
l Quadratic Regression
l Quadrilateralsl Quartilesl Quotient of
Powers Property
R
l Radian Measure of an Angle
l Radian to Degree Measure
l Radical Expressions
l Radical Equations, Solving
l Radius of Convergence
l Radius l Random Samplingl Random Variablel Range (of Data)l Rate of Changel Range (of a
Function) l Rate-Time-
Distance Problems
l Rates and Ratiosl Ratesl Rational
Equations, Solving
l Rational Exponents
l Rational Functions
l Rational Expressions
l Rational Numbers l Rationalizing the
Denominator l Ratiosl Rays l Real Numbersl Real Zero of a
Function l Reciprocals
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Review Topics
l Rectanglel Recursive
Sequencel Reduced Radical
Forml Reducing
Fractionsl Reference Angle l Reflections l Reflexive,
Symmetric, Transitive Properties
l Relationsl Relatively Prime
Numbersl Remainder
Theoreml Repeated Zeros l Repeating
Decimals l Rhombusl Right Triangle
Congruencel Rootl Rotational
Symmetryl Rotationsl Rounding
Decimals l Rounding
Numbers
S
l Sample Spacel SAS Inequalityl SAS Postulatel Scalar
Multiplication of Matrices
l Scalar Multiplication of Vectors
l Scale Factor l Scalene Triangle l Scatter Plots l Scientific
Notation l Sectorsl Segmentsl Segment Addition
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Review Topics
Postulatel Sequence l Series l Setsl Side-Angle-Side
Similarityl Side-Side-Side
Similarityl Sigma Notation
of a Seriesl Significant Digitsl Similar Figures l Similar Trianglesl Simple Interestl Simplest Form (of
a Fraction) l Simplifying
Absolute Value Expressions
l Simplifying Expressions
l Simplifying Radical Expressions
l Simplifying Rational Expressions
l Simplifying Square Roots
l Sine Functionl Sinel Slant Height of a
Pyramidl Slopel Slope-Intercept
Form (of a Linear Equation)
l Solution Setsl Solving Absolute
Value Equationsl Solving Equationsl Solving
Exponential Equations using Logarithms
l Solving Exponential Equations
l Solving Linear Equations
l Solving Linear
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Review Topics
Inequalities l Solving
Logarithmic Equations
l Solving Matrix Equations
l Solving Multi-Step Linear Equations with Decimals
l Solving Multi-Step Linear Equations with Fractions
l Solving Multi-Step Linear Equations
l Solving One-Step Linear Equations with Decimals
l Solving One-Step Linear Equations with Fractions
l Solving One-Step Linear Equations with Mixed Numbers
l Solving One-Step Linear Equations
l Solving One-Step Linear Inequalities
l Solving Problems with Vectors
l Solving Quadratic Equations by Square Roots
l Solving Quadratic Equations using Factoring
l Solving Radical Equations
l Solving Rational Equations
l Solving Systems of Linear Equations Using Elimination
l Solving Systems of Linear Equations Using Graphing
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Review Topics
l Solving Systems of Linear Equations
l Solving Systems of Linear Equations Using Matrices
l Solving Systems of Linear Equations Using Substitution
l Solving Trigonometric Equations using Algebraic Method
l Solving Trigonometric Equations using Trigonometric Identities
l Solving Two-Step Linear Equations with Decimals
l Solving Two-Step Linear Equations with Fractions
l Solving Two-Step Linear Equations
l Spherel Squarel Squares
Circumscribed by Circles
l Square Matrix l Square of a
Binomial l Square Rootsl SSS Inequalityl SSS Postulatel SSS Similarityl Standard
Deviationl Standard Form
(of a Linear Equation)
l Statistical Questions
l Stem-and-Leaf Plots
l Step Functionl Subsets
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Review Topics
l Sum and Difference Identities
l Sum of the First n Terms of a Geometric Sequence
l Sum of the First n Terms of an Arithmetic Sequence
l Sum of the First n Terms of a Series
l Sum-to-Product and Product-to-Sum Identities
l Supplementary Angles
l Supplement Postulate
l Surface Areal Surface Area of a
Conel Surface Area of a
Cylinderl Surface Area of a
Prisml Surface Area of a
Pyramidl Surface Area of a
Spherel Symbolic Logicl Systems of Linear
Equations l Systems of Linear
Inequalities
T
l Tangent Functionl Tangent of an
Anglel Tangent Linel Tangent to a
Circlel Taylor Seriesl Term of a
Sequencel Terminating and
Repeating Decimals
l Tessellations
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Review Topics
l 30-60-90 Triangles
l Traceability of Graphs
l Transcendental Numbers
l Transformation of Graphs using Matrices - Translation
l Transformation of Graphs using Matrices - Dilation
l Transformation of Graphs using Matrices - Reflection
l Transformation of Graphs using Matrices - Rotations
l Transformation of Graphs using Matrices
l Transformations of Functions
l Transformations of Graphs
l Translations l Transversall Trapezoidl Triangle Angle
Bisector Theoreml Triangle
Inequality Theorem
l Triangle Midsegment Theorem
l Triangle Proportionality Theorem
l Trianglesl Trigonometric
Functionsl Trigonometric
Identitiesl Trigonometric
Ratios l Trigonometric
Values for
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Common Anglesl Two Dimensional
Views of Three Dimensional Objects
U
l Unit Analysisl Unit Rates l Unit Vectors l Using Expected
Values to Make Decisions
l Using Probabilities to Make Fair Decisions
V
l Values of Trigonometric Ratios for Standard Angles
l Variablesl Variancel Vectorsl Venn Diagrams l Vertex of a
Parabola l Vertical Angles l Vertical Angles
Theorem l Vertical Line Test l Vertical Linesl Volume of a Conel Volume of a
Cylinderl Volume of a Prisml Volume of a
Pyramidl Volume of a
Spherel Volume
W
l Whole Numbersl Word Problems:
Area and Perimeter of a Rectangle
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Review Topics
l Word Problems: Equivalent Expressions
l Word Problems: Expected Value
l Word Problems Inverse Variation
l Word Problems Involving Decimals
l Word Problems Involving Integers
l Word Problems Involving One Step Linear Equations
l Word Problems Involving the Distributive Property
l Word Problems Involving Percent
l Word Problems Involving Width, Length and Area
l Word Problems: Linear Models
l Word Problems: Quadratic Equations
l Word Problems: Ratios
l Word Problems: Work and Workers
l Word Problems l Writing Number
Patterns in Function Notation
l Writing Systems of Linear Equations from Word Problems
X
l x-Intercepts of a Quadratic Function
l x-Intercepts
Y
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l y-Intercepts a2 level
l y-Interceptsl Yang Huis
Triangle
Z
l Zero Exponents l Zero to the Power
of Zero l Zero Product
Property
l Feedback
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