Rational Inequalities
•Download as PPTX, PDF•
0 likes•28 views
This document discusses solving rational inequalities and the wavy curve method. It explains that rational inequalities involve fractions with real number coefficients and natural number exponents. It provides the example of (x-1)(x-2)>0 and explains the solution set is between the critical points of x=1 and x=2. It also discusses using the wavy curve method to determine the solution sets of rational inequalities, which involves drawing a wavy curve above the number line that passes through critical points and indicates whether sections are positive or negative.
Report
Share
Report
Share
![Solving Rational inequalities
• (or >0, ≥0, ≤0) where a1, a2, …… am, b1,b2,…… bm are
real numbers and n1,n2,…………nm,p1,p2,………pm are natural numbers.
• On real number line (x-1)(x-2)>0
• For the above inequality we can see x=1 and 2 are critical points.
• Solution set is x ϵ (-α,1) U (2,α)
• Similarly
+ (x-1)(x-2) > 0, x ϵ (1,2)
+ (x-1)(x-2) ≥ 0, x ϵ (-α,1] U [2,α)
+ (x-1)(x-2) ≤ 0, x ϵ [1,2]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
Linear-Algebra.ppt
The document discusses solving systems of linear algebraic equations. It describes representing a system of n equations with n unknowns in matrix/vector form as Ax=b. It covers Gaussian elimination methods for solving such systems, including elementary row operations, partial pivoting to avoid division by zero, and LU decomposition. It also provides an example of solving a system in MATLAB using the backslash operator or LU decomposition with the lu command.
Linear-Algebra.ppt
The document discusses solving systems of linear algebraic equations. It describes representing a system of n equations with n unknowns in matrix/vector form as Ax=b. It covers Gaussian elimination methods for solving such systems, including elementary row operations, partial pivoting to avoid division by zero, and LU decomposition. It also provides an example of solving a system in MATLAB using the backslash operator or LU decomposition with the lu command.
07-Convolution.pptx signal spectra and signal processing
The document discusses convolution, which is a mathematical operation used to describe the output of a linear time-invariant system when its input is another function. Convolution combines two sequences to produce a third sequence. It has properties like commutativity and associativity. To perform a convolution, the sequences can be directly evaluated as a summation, plotted graphically and multiplied/summed, or calculated using a slide rule method by shifting and multiplying corresponding terms.
1638 vector quantities
This document discusses vector algebra concepts including scalar and vector quantities, vector operations like addition and subtraction, and different coordinate systems used to represent vectors such as Cartesian, cylindrical, and spherical coordinates. Key topics covered include defining a vector using basis vectors in a coordinate system, calculating the magnitude and direction of a vector, adding and subtracting vectors using graphical representations, computing the dot and cross products of vectors, and converting between Cartesian, cylindrical, and spherical coordinate representations of vectors.
Advanced Search Techniques
The document discusses different search techniques including binary search, bisection, and ternary search. It provides descriptions of binary search and gives iterative and recursive pseudocode implementations. It also discusses the logarithmic worst case performance of binary search. Ternary search is introduced as a technique for finding the minimum or maximum of a unimodal function. Bisection is described as dividing elements into two parts and sorting one part to enable binary search.
Computer graphics - bresenham line drawing algorithm
You know that DDA algorithm is an incremental scan conversion method which performs calculations at each step using the results from the preceding step. Here we are going to discover an accurate and efficient raster line generating algorithm, the Bresenham's line-drawing algorithm.
This algorithm was developed by Jack E. Bresenham in 1962 at IBM.
The function given below handles all lines and implements the complete Bresenham's algorithm.
function line(x0, x1, y0, y1)
boolean steep := abs(y1 - y0) > abs(x1 - x0)
if steep then
swap(x0, y0)
swap(x1, y1)
if x0 > x1 then
swap(x0, x1)
swap(y0, y1)
int deltax := x1 - x0
int deltay := abs(y1 - y0)
real error := 0
real deltaerr := deltay / deltax
int y := y0
if y0 < y1 then ystep := 1 else ystep := -1
for x from x0 to x1
if steep then plot(y,x) else plot(x,y)
error := error + deltaerr
if error ? 0.5
y := y + ystep
error := error - 1.0
linearequns-classx-180912070018.pdf
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a single linear equation represents a straight line and can have infinitely many solution pairs (x,y). The document also discusses how two linear equations can have a unique solution if their lines intersect, no solution if the lines are parallel, or infinitely many solutions if the lines are coincident. Finally, it presents different algebraic methods for solving systems of two linear equations, including substitution, elimination of coefficients, and cross-multiplication.
CLASS X MATHS LINEAR EQUATIONS
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a linear equation can have infinitely many solutions (x,y value pairs) that satisfy the equation, and these solutions lie on a straight line. The document provides an example of a single linear equation and shows its graph on the Cartesian plane. It also discusses systems of two linear equations, explaining that their solutions occur where the lines intersect. The document covers various algebraic methods for solving systems of linear equations, including elimination by substitution or equating coefficients, and solving by cross multiplication. It provides examples to illustrate these solution
Recommended
Linear-Algebra.ppt
The document discusses solving systems of linear algebraic equations. It describes representing a system of n equations with n unknowns in matrix/vector form as Ax=b. It covers Gaussian elimination methods for solving such systems, including elementary row operations, partial pivoting to avoid division by zero, and LU decomposition. It also provides an example of solving a system in MATLAB using the backslash operator or LU decomposition with the lu command.
Linear-Algebra.ppt
The document discusses solving systems of linear algebraic equations. It describes representing a system of n equations with n unknowns in matrix/vector form as Ax=b. It covers Gaussian elimination methods for solving such systems, including elementary row operations, partial pivoting to avoid division by zero, and LU decomposition. It also provides an example of solving a system in MATLAB using the backslash operator or LU decomposition with the lu command.
07-Convolution.pptx signal spectra and signal processing
The document discusses convolution, which is a mathematical operation used to describe the output of a linear time-invariant system when its input is another function. Convolution combines two sequences to produce a third sequence. It has properties like commutativity and associativity. To perform a convolution, the sequences can be directly evaluated as a summation, plotted graphically and multiplied/summed, or calculated using a slide rule method by shifting and multiplying corresponding terms.
1638 vector quantities
This document discusses vector algebra concepts including scalar and vector quantities, vector operations like addition and subtraction, and different coordinate systems used to represent vectors such as Cartesian, cylindrical, and spherical coordinates. Key topics covered include defining a vector using basis vectors in a coordinate system, calculating the magnitude and direction of a vector, adding and subtracting vectors using graphical representations, computing the dot and cross products of vectors, and converting between Cartesian, cylindrical, and spherical coordinate representations of vectors.
Advanced Search Techniques
The document discusses different search techniques including binary search, bisection, and ternary search. It provides descriptions of binary search and gives iterative and recursive pseudocode implementations. It also discusses the logarithmic worst case performance of binary search. Ternary search is introduced as a technique for finding the minimum or maximum of a unimodal function. Bisection is described as dividing elements into two parts and sorting one part to enable binary search.
Computer graphics - bresenham line drawing algorithm
You know that DDA algorithm is an incremental scan conversion method which performs calculations at each step using the results from the preceding step. Here we are going to discover an accurate and efficient raster line generating algorithm, the Bresenham's line-drawing algorithm.
This algorithm was developed by Jack E. Bresenham in 1962 at IBM.
The function given below handles all lines and implements the complete Bresenham's algorithm.
function line(x0, x1, y0, y1)
boolean steep := abs(y1 - y0) > abs(x1 - x0)
if steep then
swap(x0, y0)
swap(x1, y1)
if x0 > x1 then
swap(x0, x1)
swap(y0, y1)
int deltax := x1 - x0
int deltay := abs(y1 - y0)
real error := 0
real deltaerr := deltay / deltax
int y := y0
if y0 < y1 then ystep := 1 else ystep := -1
for x from x0 to x1
if steep then plot(y,x) else plot(x,y)
error := error + deltaerr
if error ? 0.5
y := y + ystep
error := error - 1.0
linearequns-classx-180912070018.pdf
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a single linear equation represents a straight line and can have infinitely many solution pairs (x,y). The document also discusses how two linear equations can have a unique solution if their lines intersect, no solution if the lines are parallel, or infinitely many solutions if the lines are coincident. Finally, it presents different algebraic methods for solving systems of two linear equations, including substitution, elimination of coefficients, and cross-multiplication.
CLASS X MATHS LINEAR EQUATIONS
This document discusses linear equations in two variables. It begins by presenting the general form of a linear equation as ax + by + c = 0, where a, b, and c are real numbers. It then explains that a linear equation can have infinitely many solutions (x,y value pairs) that satisfy the equation, and these solutions lie on a straight line. The document provides an example of a single linear equation and shows its graph on the Cartesian plane. It also discusses systems of two linear equations, explaining that their solutions occur where the lines intersect. The document covers various algebraic methods for solving systems of linear equations, including elimination by substitution or equating coefficients, and solving by cross multiplication. It provides examples to illustrate these solution
Ch9-Gauss_Elimination4.pdf
This document provides information about solving systems of linear algebraic equations using Gauss elimination. It begins by defining linear equations and systems of linear equations. It then discusses some non-computer methods for solving systems of equations, including graphical methods, Cramer's rule, and the method of elimination. The majority of the document focuses on explaining Gauss elimination, including forward elimination to obtain an upper triangular system and back substitution to solve for the unknowns. It provides an example of using the naive Gauss elimination method to solve a system of 4 equations with 4 unknowns step-by-step.
Matlab lab manual
The document describes three numerical methods for finding the roots or solutions of equations: the bisection method, Newton's method for single variable equations, and Newton's method for systems of nonlinear equations.
The bisection method works by repeatedly bisecting the interval within which a root is known to exist, narrowing in on the root through iterative halving. Newton's method approximates the function with its tangent line to find a better root estimate with each iteration. For systems of equations, Newton's method involves calculating the Jacobian matrix and solving a system of linear equations at each step to update the solution estimate. Examples are provided to illustrate each method.
Applied Algorithms and Structures week999
The document discusses algorithms and data structures. It begins with two quotes about programming and algorithms. It then provides pseudocode for naive and optimized recursive Fibonacci algorithms, as well as an iterative dynamic programming version. It also covers dynamic programming approaches for calculating Fibonacci numbers, Catalan numbers, the chessboard traversal problem, the rod cutting problem, longest common subsequence, and assembly line traversal. The key ideas are introducing dynamic programming techniques like memoization and bottom-up iteration to improve the time complexity of recursive algorithms from exponential to polynomial.
Nbhm m. a. and m.sc. scholarship test september 20, 2014 with answer key
The document contains instructions for a scholarship test consisting of 30 questions divided into 3 sections: Algebra, Analysis, and Miscellaneous. It provides guidance on answering questions, allowed resources, and key definitions and notations used in the test. Candidates are instructed to answer each question in the supplied answer booklet and not on the question paper itself. Calculators are not permitted.
Mathematics ppt.pptx
1) The document provides information about linear equations in two variables including the general form of a linear equation, single linear equations, systems of two linear equations, conditions for common solutions, and methods to solve systems of linear equations algebraically.
2) Examples are provided to illustrate graphing single linear equations, finding common solutions to systems of two linear equations, and solving systems using elimination and cross-multiplication methods.
3) Key methods for solving systems of linear equations discussed include elimination by substitution or equating coefficients, and cross-multiplication. Conditions for common solutions depend on whether lines intersect, are parallel, or are coincident.
Unit 1 Operation on signals
1. The document discusses various operations that can be performed on signals including time reversal, time shifting, time scaling, amplitude scaling, signal addition, and signal multiplication.
2. Examples are provided to demonstrate how to graphically represent signals and how the different operations change the signals.
3. Key steps are outlined for performing each operation including reversing the time axis, delaying or advancing signals, compressing or expanding the time axis, amplifying or attenuating signal amplitude, adding or multiplying signal values.
Classification of systems : Part 2
The document discusses various types of systems including:
- Continuous and discrete time systems
- Linear and non-linear systems
It provides examples of linear systems such as electrical circuits and examples of non-linear systems such as amplitude modulation. It also discusses properties of linear systems including additivity and homogeneity. Stable and unstable systems are defined based on whether bounded inputs produce bounded outputs. Several examples are provided to determine if systems are linear, time-invariant, causal, stable or not.
22 01 2014_03_23_31_eee_formula_sheet_final
This document provides information about a formula sheet published by the Institute of Engineering Studies in Bangalore, India. The formula sheet contains formulas and concepts for 12 engineering topics from mathematics to power electronics. It also lists contact information for the publisher and provides details about online and offline services available to exam preparation aspirants, including classroom coaching, study materials, practice question books, and access to an online test portal.
ゲーム理論 BASIC 演習74 補足 -コアの存在, 線型計画問題-
The document discusses three linear programming problems to minimize the sum of variables x1, x2, and x3 subject to different constraints on the values of subsets. The first problem has the constraints v({1,2})=2, v({1,3})=3, v({2,3})=2 and has an optimal value of 7/2. The second problem changes the constraint v({1,3})=4 and has an optimal value of 4. The third problem changes the constraint v({1,2})=3 and has an optimal value of 9/2.
Linear Algebra and Matlab tutorial
- The document provides an introduction to linear algebra and MATLAB. It discusses various linear algebra concepts like vectors, matrices, tensors, and operations on them.
- It then covers key MATLAB topics - basic data types, vector and matrix operations, control flow, plotting, and writing efficient code.
- The document emphasizes how linear algebra and MATLAB are closely related and commonly used together in applications like image and signal processing.
Computational Method for Engineers: Solving a system of Linear Equations
System of Linear equations are solved using different numerical methods, mainly categorised under direct and indirect with MATLAB Code.
NN-Ch2.PDF
1. The document introduces neural networks and concepts such as linear separability.
2. A simple single-layer neural network can classify patterns into two classes if the classes are linearly separable by a hyperplane defined by the network's weights and bias.
3. Linearly separable problems can be solved by this network, but linearly inseparable problems require more complex network architectures.
Computation in Real Closed Infinitesimal and Transcendental Extensions of the...
Transcendental Numbers
Power series
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
Sistempertidaksamaanduavariabel2122
The document discusses systems of two variable equations and inequalities. It defines a system of two variable equations as a collection of two or more equations involving two variables (linear-linear, linear-quadratic, quadratic-quadratic) whose solutions are points satisfying the equations. The solution graph is the intersection of the equations. It also defines systems of two variable inequalities and explains how to graphically represent the solution sets. Examples of solving systems using graphical and substitution methods are provided.
Linear regression, costs & gradient descent
Linear Regression is a predictive model to map the relation between the dependent variable and one or more independent variables.
CS767_Lecture_04.pptx
The document discusses perceptrons and artificial neural networks, including describing the architecture and learning algorithm of a simple perceptron for pattern classification, providing an example of how the perceptron learns from labeled training data to determine the weights and bias that allow it to classify new input patterns, and assigning exercises to modify the perceptron code provided to classify different input data and analyze its convergence over training epochs.
Arithmetic Progressions and the Construction of Doubly Even Magic Squares (Fo...
Arithmetic Progressions and the Construction of Doubly Even Magic Squares (Fo...Lossian Barbosa Bacelar Miranda
In this brief article we present an interesting use of arithmetic progressions to construct doubly even magic squares.Detailed Description on Cross Entropy Loss Function
The document discusses cross entropy loss function which is commonly used in classification problems. It derives the theoretical basis for cross entropy by formulating it as minimizing the cross entropy between the predicted probabilities and true labels. For binary classification problems, cross entropy is shown to be equivalent to maximizing the likelihood of the training data which can be written as minimizing the binary cross entropy. This concept is extended to multiclass classification problems by defining the prediction as a probability distribution over classes and label as a one-hot encoding.
Optimum Engineering Design - Day 2b. Classical Optimization methods
This document provides an overview of an optimization methods course, including its objectives, prerequisites, and materials. The course covers topics such as linear programming, nonlinear programming, and mixed integer programming problems. It also includes mathematical preliminaries on topics like convex sets and functions, gradients, Hessians, and Taylor series expansions. Methods for solving systems of linear equations and examples are presented.
Trigonometry formulas | JEE Mathematics in Tamil | Season 0, Episode 7 | Fund...
This document discusses trigonometry ratios and basic formulae based on the NCERT syllabus for JEE Maths from basic to advanced levels in Tamil. It covers the range of values for trigonometric ratios like sine, cosine, tangent, cosecant, secant and cotangent from 0 to 90 degrees and introduces basic trigonometric formulae. The document is labeled as episode 8 and focuses on trigonometry problems.
Intervals
This document provides definitions and examples of different types of intervals in mathematics including open intervals, closed intervals, and semi-closed or semi-open intervals. It defines open intervals as not including the endpoints, closed intervals as including both endpoints when finite, and semi-closed intervals as including one endpoint but not the other. Examples are given for each type of interval using inequalities to represent membership in the set.
More Related Content
Similar to Rational Inequalities
Ch9-Gauss_Elimination4.pdf
This document provides information about solving systems of linear algebraic equations using Gauss elimination. It begins by defining linear equations and systems of linear equations. It then discusses some non-computer methods for solving systems of equations, including graphical methods, Cramer's rule, and the method of elimination. The majority of the document focuses on explaining Gauss elimination, including forward elimination to obtain an upper triangular system and back substitution to solve for the unknowns. It provides an example of using the naive Gauss elimination method to solve a system of 4 equations with 4 unknowns step-by-step.
Matlab lab manual
The document describes three numerical methods for finding the roots or solutions of equations: the bisection method, Newton's method for single variable equations, and Newton's method for systems of nonlinear equations.
The bisection method works by repeatedly bisecting the interval within which a root is known to exist, narrowing in on the root through iterative halving. Newton's method approximates the function with its tangent line to find a better root estimate with each iteration. For systems of equations, Newton's method involves calculating the Jacobian matrix and solving a system of linear equations at each step to update the solution estimate. Examples are provided to illustrate each method.
Applied Algorithms and Structures week999
The document discusses algorithms and data structures. It begins with two quotes about programming and algorithms. It then provides pseudocode for naive and optimized recursive Fibonacci algorithms, as well as an iterative dynamic programming version. It also covers dynamic programming approaches for calculating Fibonacci numbers, Catalan numbers, the chessboard traversal problem, the rod cutting problem, longest common subsequence, and assembly line traversal. The key ideas are introducing dynamic programming techniques like memoization and bottom-up iteration to improve the time complexity of recursive algorithms from exponential to polynomial.
Nbhm m. a. and m.sc. scholarship test september 20, 2014 with answer key
The document contains instructions for a scholarship test consisting of 30 questions divided into 3 sections: Algebra, Analysis, and Miscellaneous. It provides guidance on answering questions, allowed resources, and key definitions and notations used in the test. Candidates are instructed to answer each question in the supplied answer booklet and not on the question paper itself. Calculators are not permitted.
Mathematics ppt.pptx
1) The document provides information about linear equations in two variables including the general form of a linear equation, single linear equations, systems of two linear equations, conditions for common solutions, and methods to solve systems of linear equations algebraically.
2) Examples are provided to illustrate graphing single linear equations, finding common solutions to systems of two linear equations, and solving systems using elimination and cross-multiplication methods.
3) Key methods for solving systems of linear equations discussed include elimination by substitution or equating coefficients, and cross-multiplication. Conditions for common solutions depend on whether lines intersect, are parallel, or are coincident.
Unit 1 Operation on signals
1. The document discusses various operations that can be performed on signals including time reversal, time shifting, time scaling, amplitude scaling, signal addition, and signal multiplication.
2. Examples are provided to demonstrate how to graphically represent signals and how the different operations change the signals.
3. Key steps are outlined for performing each operation including reversing the time axis, delaying or advancing signals, compressing or expanding the time axis, amplifying or attenuating signal amplitude, adding or multiplying signal values.
Classification of systems : Part 2
The document discusses various types of systems including:
- Continuous and discrete time systems
- Linear and non-linear systems
It provides examples of linear systems such as electrical circuits and examples of non-linear systems such as amplitude modulation. It also discusses properties of linear systems including additivity and homogeneity. Stable and unstable systems are defined based on whether bounded inputs produce bounded outputs. Several examples are provided to determine if systems are linear, time-invariant, causal, stable or not.
22 01 2014_03_23_31_eee_formula_sheet_final
This document provides information about a formula sheet published by the Institute of Engineering Studies in Bangalore, India. The formula sheet contains formulas and concepts for 12 engineering topics from mathematics to power electronics. It also lists contact information for the publisher and provides details about online and offline services available to exam preparation aspirants, including classroom coaching, study materials, practice question books, and access to an online test portal.
ゲーム理論 BASIC 演習74 補足 -コアの存在, 線型計画問題-
The document discusses three linear programming problems to minimize the sum of variables x1, x2, and x3 subject to different constraints on the values of subsets. The first problem has the constraints v({1,2})=2, v({1,3})=3, v({2,3})=2 and has an optimal value of 7/2. The second problem changes the constraint v({1,3})=4 and has an optimal value of 4. The third problem changes the constraint v({1,2})=3 and has an optimal value of 9/2.
Linear Algebra and Matlab tutorial
- The document provides an introduction to linear algebra and MATLAB. It discusses various linear algebra concepts like vectors, matrices, tensors, and operations on them.
- It then covers key MATLAB topics - basic data types, vector and matrix operations, control flow, plotting, and writing efficient code.
- The document emphasizes how linear algebra and MATLAB are closely related and commonly used together in applications like image and signal processing.
Computational Method for Engineers: Solving a system of Linear Equations
System of Linear equations are solved using different numerical methods, mainly categorised under direct and indirect with MATLAB Code.
NN-Ch2.PDF
1. The document introduces neural networks and concepts such as linear separability.
2. A simple single-layer neural network can classify patterns into two classes if the classes are linearly separable by a hyperplane defined by the network's weights and bias.
3. Linearly separable problems can be solved by this network, but linearly inseparable problems require more complex network architectures.
Computation in Real Closed Infinitesimal and Transcendental Extensions of the...
Transcendental Numbers
Power series
Series solution of ordinary differential equation
advance engineering mathematics
The power series method is the standard method for solving linear ODEs with variable
coefficients. It gives solutions in the form of power series. These series can be used for computing values, graphing curves, proving formulas, and exploring properties of
solutions, as we shall see.
Sistempertidaksamaanduavariabel2122
The document discusses systems of two variable equations and inequalities. It defines a system of two variable equations as a collection of two or more equations involving two variables (linear-linear, linear-quadratic, quadratic-quadratic) whose solutions are points satisfying the equations. The solution graph is the intersection of the equations. It also defines systems of two variable inequalities and explains how to graphically represent the solution sets. Examples of solving systems using graphical and substitution methods are provided.
Linear regression, costs & gradient descent
Linear Regression is a predictive model to map the relation between the dependent variable and one or more independent variables.
CS767_Lecture_04.pptx
The document discusses perceptrons and artificial neural networks, including describing the architecture and learning algorithm of a simple perceptron for pattern classification, providing an example of how the perceptron learns from labeled training data to determine the weights and bias that allow it to classify new input patterns, and assigning exercises to modify the perceptron code provided to classify different input data and analyze its convergence over training epochs.
Arithmetic Progressions and the Construction of Doubly Even Magic Squares (Fo...
Arithmetic Progressions and the Construction of Doubly Even Magic Squares (Fo...Lossian Barbosa Bacelar Miranda
In this brief article we present an interesting use of arithmetic progressions to construct doubly even magic squares.Detailed Description on Cross Entropy Loss Function
The document discusses cross entropy loss function which is commonly used in classification problems. It derives the theoretical basis for cross entropy by formulating it as minimizing the cross entropy between the predicted probabilities and true labels. For binary classification problems, cross entropy is shown to be equivalent to maximizing the likelihood of the training data which can be written as minimizing the binary cross entropy. This concept is extended to multiclass classification problems by defining the prediction as a probability distribution over classes and label as a one-hot encoding.
Optimum Engineering Design - Day 2b. Classical Optimization methods
This document provides an overview of an optimization methods course, including its objectives, prerequisites, and materials. The course covers topics such as linear programming, nonlinear programming, and mixed integer programming problems. It also includes mathematical preliminaries on topics like convex sets and functions, gradients, Hessians, and Taylor series expansions. Methods for solving systems of linear equations and examples are presented.
Similar to Rational Inequalities (20)
Nbhm m. a. and m.sc. scholarship test september 20, 2014 with answer key
Nbhm m. a. and m.sc. scholarship test september 20, 2014 with answer key
Computational Method for Engineers: Solving a system of Linear Equations
Computational Method for Engineers: Solving a system of Linear Equations
Computation in Real Closed Infinitesimal and Transcendental Extensions of the...
Computation in Real Closed Infinitesimal and Transcendental Extensions of the...
Arithmetic Progressions and the Construction of Doubly Even Magic Squares (Fo...
Arithmetic Progressions and the Construction of Doubly Even Magic Squares (Fo...
Detailed Description on Cross Entropy Loss Function
Detailed Description on Cross Entropy Loss Function
Optimum Engineering Design - Day 2b. Classical Optimization methods
Optimum Engineering Design - Day 2b. Classical Optimization methods
More from Ahamed Yoonus S
Trigonometry formulas | JEE Mathematics in Tamil | Season 0, Episode 7 | Fund...
This document discusses trigonometry ratios and basic formulae based on the NCERT syllabus for JEE Maths from basic to advanced levels in Tamil. It covers the range of values for trigonometric ratios like sine, cosine, tangent, cosecant, secant and cotangent from 0 to 90 degrees and introduces basic trigonometric formulae. The document is labeled as episode 8 and focuses on trigonometry problems.
Intervals
This document provides definitions and examples of different types of intervals in mathematics including open intervals, closed intervals, and semi-closed or semi-open intervals. It defines open intervals as not including the endpoints, closed intervals as including both endpoints when finite, and semi-closed intervals as including one endpoint but not the other. Examples are given for each type of interval using inequalities to represent membership in the set.
Ratio and Proportions
This document provides an overview of ratios and proportions based on the NCERT syllabus for JEE Maths B2A. It defines ratio as quantities of the same kind represented by A:B or the fraction A/B, and defines proportions as equal ratios such that a:b = c:d. It lists important properties of proportions including the product of extremes equals the product of means and inverse, alternate, componendo, and dividend properties. It then transitions to solving equations and inequalities related to ratios and proportions.
JEE Mathematics B2A in Tamil | Season 0, Episode 3 | Fundamentals of Mathemat...
The document provides information about polynomials including:
1) The definition of a polynomial as an expression involving terms of various integer powers of x with the highest power term having a non-zero coefficient.
2) The remainder theorem which states that if a polynomial p(x) is divided by a binomial x-a, the remainder is equal to p(a).
3) The factor theorem which states that if a polynomial f(x) is equal to 0 at x=a, then (x-a) is a factor of the polynomial.
JEE Mathematics B2A in Tamil | Season 0, Episode 2 | Fundamentals of Mathemat...
This document provides formulas and examples related to polynomials for JEE math preparation. It includes:
1) Formulas for expanding and factorizing polynomial expressions like (a+b)2, (a-b)3, a3 + b3, etc.
2) Examples of problems involving polynomials like finding the value of expressions given certain conditions.
3) A problem asking to find real numbers a and b given their sum in a quadratic equation.
4) A problem factorizing the polynomial x4 + 5x2 + 9.
5) An example of simplifying a complex polynomial expression.
Fundamental of Mathematics - Number system
This document provides an overview of different number systems including natural numbers, integers, rational numbers, and irrational numbers. It defines key terms such as prime numbers, composite numbers, co-prime numbers, and twin prime numbers. Examples are given for each type of number. The document also discusses properties of integers, rational numbers, and irrational numbers. It provides questions to test understanding of different number concepts covered.
Force and pressure introduction - Yoonus
Force is a push or pull that causes motion or changes the motion of an object. Forces act between two interacting objects. The motion of objects is caused by the action of a force - for example, a magnet attracts iron due to the force of attraction. A push is when one object exerts a force to move another object away, while a pull is when an object exerts a force to move another object towards itself. Forces can add up if acting in the same direction or subtract if acting in opposite directions. The strength and direction of a force determines its effect.
Force and pressure - state of motion
The document discusses the effects that forces can have on objects. It states that a force can change an object's speed by either increasing or decreasing it depending on whether the force is applied in the same or opposite direction of the object's motion. A force can also change an object's direction of motion or shape. However, a force does not always result in a change in an object's state of motion, as experience shows that sometimes applying a force does not change an object's motion. Forces are also categorized as either contact forces, which require physical contact, or non-contact forces, which act over a distance.
Force and pressure - Pressure
Pressure is defined as force per unit area. It is exerted by both liquids and gases on the walls of their containers. Atmospheric pressure is caused by the weight of air in the atmosphere and it is very large, as demonstrated by the force needed to remove a sucker from a surface or the inability of eight horses to pull apart evacuated hemispheres. The human body can withstand this pressure because internal and external pressures are equal.
Force and pressure - kinds of forces
Force can be defined as a push or pull that causes an object to be in motion or attempt to change its motion. There are several different types of forces including contact forces like muscular and frictional forces, and non-contact forces like magnetic, electrostatic, and gravitational forces. Muscular force allows humans and animals to perform physical tasks, while friction always acts opposite to an object's motion and eventually stops rolling objects. Magnetic force causes attraction or repulsion between magnets without contact, and electrostatic force allows charged objects to influence each other at a distance. Gravitational force is responsible for all objects being pulled toward Earth and other massive bodies.
physical and chemical changes | Introduction
This is 7th standard chapter 6 cbse science part.
It contains physical change part only.
Check out other slides for more topics.
physical and chemical changes | Physical change
This is 7th standard chapter 6 cbse science part.
It contains physical change part only.
Check out other slides for more topics.
Acid, Bases and salts | Indicators
This is 7th standard science topics.
Chapter 5
Acid bases and salts.
In that the sub category is Indicators
Current electricity | Fuse, Series and parallel
This slide is part 3 of current electricity. We have uploaded lot more before. check that out and then study well.
Thank you.
Electricity - Resistors in series and parallel
Resistors in series and parallel.
For more other electricity chapters see our other files.
Electricity 10th standard science - physics.
Electricity
Electricity. Ohms law. and all other 10th standard physics concepts part 1.
see other parts for more information.
Current electricity Part -1
10th standard science chapter Tamil Nadu state board syllabus - chapter 12.
This is just a part 1 of current electricity chapter.
The next parts are available here, just check it out and learn Electricity.
Practical Learning and assessment
Advantages and disadvantages in practical learning method and practical assessment.
Why there is a practical learning in education?
B.Ed., course paper.
Role of ICT in examination
This document discusses the role of information and communication technology (ICT) in educational assessment. It states that ICT has led to changes in assessment by allowing for new formats like computer-based testing and higher-order learning outcomes. ICT can be used to support both formative and summative assessment. Examples of ICT tools for assessment include online testing platforms, survey tools, and software for creating and scoring tests. ICT also facilitates project-based assessment and helps support students with disabilities.
Details about CUCET entrance exam
CUCET entrance exam is for entering into Different central universities around India.
This ppt is specially based on CUTN (Central University of TamilNadu)
More from Ahamed Yoonus S (20)
Trigonometry formulas | JEE Mathematics in Tamil | Season 0, Episode 7 | Fund...
Trigonometry formulas | JEE Mathematics in Tamil | Season 0, Episode 7 | Fund...
JEE Mathematics B2A in Tamil | Season 0, Episode 3 | Fundamentals of Mathemat...
JEE Mathematics B2A in Tamil | Season 0, Episode 3 | Fundamentals of Mathemat...
JEE Mathematics B2A in Tamil | Season 0, Episode 2 | Fundamentals of Mathemat...
JEE Mathematics B2A in Tamil | Season 0, Episode 2 | Fundamentals of Mathemat...
Recently uploaded
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS Template 2023-2024 by: Irene S. Rueco
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
-------------------------------------------------------------------------------
Find out more about ISO training and certification services
Training: ISO/IEC 27001 Information Security Management System - EN | PECB
ISO/IEC 42001 Artificial Intelligence Management System - EN | PECB
General Data Protection Regulation (GDPR) - Training Courses - EN | PECB
Webinars: https://pecb.com/webinars
Article: https://pecb.com/article
-------------------------------------------------------------------------------
For more information about PECB:
Website: https://pecb.com/
LinkedIn: https://www.linkedin.com/company/pecb/
Facebook: https://www.facebook.com/PECBInternational/
Slideshare: http://www.slideshare.net/PECBCERTIFICATION
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
How to Manage Your Lost Opportunities in Odoo 17 CRM
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar KanvariaNatural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama UniversityAkanksha trivedi rama nursing college kanpur.
Natural birth techniques are various type such as/ water birth , alexender method, hypnosis, bradley method, lamaze method etcHow to Add Chatter in the odoo 17 ERP Module
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)Academy of Science of South Africa
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Walmart Business+ and Spark Good for Nonprofits.pdf
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Hindi varnamala | hindi alphabet PPT.pdf
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
How to Setup Warehouse & Location in Odoo 17 Inventory
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
Community pharmacy- Social and preventive pharmacy UNIT 5
Covered community pharmacy topic of the subject Social and preventive pharmacy for Diploma and Bachelor of pharmacy
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...Nguyen Thanh Tu Collection
https://app.box.com/s/y977uz6bpd3af4qsebv7r9b7s21935vdRecently uploaded (20)
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Pengantar Penggunaan Flutter - Dart programming language1.pptx
Pengantar Penggunaan Flutter - Dart programming language1.pptx
How to Manage Your Lost Opportunities in Odoo 17 CRM
How to Manage Your Lost Opportunities in Odoo 17 CRM
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Natural birth techniques - Mrs.Akanksha Trivedi Rama University
Digital Artefact 1 - Tiny Home Environmental Design
Digital Artefact 1 - Tiny Home Environmental Design
South African Journal of Science: Writing with integrity workshop (2024)
South African Journal of Science: Writing with integrity workshop (2024)
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
বাংলাদেশ অর্থনৈতিক সমীক্ষা (Economic Review) ২০২৪ UJS App.pdf
Walmart Business+ and Spark Good for Nonprofits.pdf
Walmart Business+ and Spark Good for Nonprofits.pdf
How to Setup Warehouse & Location in Odoo 17 Inventory
How to Setup Warehouse & Location in Odoo 17 Inventory
Film vocab for eal 3 students: Australia the movie
Film vocab for eal 3 students: Australia the movie
Community pharmacy- Social and preventive pharmacy UNIT 5
Community pharmacy- Social and preventive pharmacy UNIT 5
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
BÀI TẬP BỔ TRỢ TIẾNG ANH 8 CẢ NĂM - GLOBAL SUCCESS - NĂM HỌC 2023-2024 (CÓ FI...
Rational Inequalities
- 1. JEE Maths B2A (Basic to Advance) in Tamil Based on NCERT syllabus
- 2. Solving Rational inequalities • (or >0, ≥0, ≤0) where a1, a2, …… am, b1,b2,…… bm are real numbers and n1,n2,…………nm,p1,p2,………pm are natural numbers. • On real number line (x-1)(x-2)>0 • For the above inequality we can see x=1 and 2 are critical points. • Solution set is x ϵ (-α,1) U (2,α) • Similarly + (x-1)(x-2) > 0, x ϵ (1,2) + (x-1)(x-2) ≥ 0, x ϵ (-α,1] U [2,α) + (x-1)(x-2) ≤ 0, x ϵ [1,2]
- 3. Wavy curve Method • Start a wave from extreme right critical point above the real number line • Which pass through all critical points making crest & trough above and below real number line • Above shows positive below shows negative • For even power, wave touches the real line and for odd power wave cuts real number line as shown in figures • For example,
- 8. Episode 7 – Trigonometry Loading…..