Absolute Value Inequalities Research Articles

Students’ commognitive conflicts in solving absolute value inequalities

The material of absolute value inequalities is considered more difficult to study. There are still many undergraduate students who still make mistakes in understanding and solving problems related to this topic. Moreover, some of them do not yet know how to read the symbols. Therefore, these issues need to be studied more thoroughly. This research was conducted to explore the undergraduate students’ discourse on absolute value inequality tasks through commognitive analysis. A qualitative approach was applied, involving 78 participants; 11 of them were further interviewed. The analysis of the absolute value inequality tasks and the interview results revealed that there were commognitive conflicts that needed to be addressed. The findings of this research showed that commognitive conflicts arose when the undergraduate students applied the definition of absolute value and determined the solution set for the absolute value inequality problems.

Analisis Kemampuan Mahasiswa Matematika FMIPA Unimed dalam Menyelesaikan Pertidaksamaan Nilai Mutlak dengan Berbantuan Python

This research aims to analyze the ability of Unimed FMIPA Mathematics students in solving absolute value inequalities with the help of the Python program. The problem faced is that students often have difficulty solving inequality problems manually. The purpose of this research is to see the effectiveness of using Python in helping to solve absolute value inequalities and how it affects understanding of mathematical concepts. The method used was qualitative research with a sample of 20 fifth semester students of the Mathematics Study Program, FMIPA, Medan State University. Students are given absolute value inequality problems to solve manually and with the help of Python. Data was collected through online tests and questionnaires using Google Form. The research results show that the majority of students feel that using Python is very helpful in solving absolute value inequalities. As many as 95% of students consider Python to be effective in making it easier to solve mathematical problems and increasing understanding of the concept of absolute value.

L₂,₁-Norm Regularized Robust and Sparse Linear Discriminant Analysis via an Alternating Direction Method of Multipliers

Recently, an absolute value inequalities discriminant analysis criterion with robustness and sparseness for supervised dimensionality reduction was studied. However, it obtains discriminant directions one by one through greedy search, which makes the sparseness of multiple discriminant directions unexplainable. In addition, it relaxes the original problem into a series of linear programming problems which makes it time consuming. In this paper, we construct a novel linear discriminant analysis with robustness and sparseness jointly through the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2,1</sub> -norm. The proposed approach obtains all the discriminant directions simultaneously, and rather than solving linear programming problems, it is solved by a more effective alternating direction method of multipliers. The effectiveness of the proposed method is supported by preliminary experimental results on two artificial datasets, some benchmark datasests and two face image datasets.

Implementing Class Wide Peer Tutoring to Enhance Students' Mathematical Reasoning Ability in Understanding Absolute Value Equations and Inequalities

This article aims to determine the effectiveness of Class Wide Peer Tutoring (CWPT) implementation to improve students' mathematical reasoning abilities. This research is experimental research with a mixed-method approach. The research sample was determined through cluster random sampling. Data were collected through mathematical reasoning tests arranged according to indicators of mathematical reasoning ability. Data were analyzed using inferential statistics with the stages of normality, homogeneity, and hypothesis testing. The result shows that the implementation of CWPT enhances students' mathematical reasoning on Equations and Inequalities of an Absolute Value. Mathematical reasoning test results show that the 10th grader score of mathematical reasoning is 76.75 (Good). The indicator of mathematical reasoning that is the most difficult for students to master is "drawing logical conclusions" with the acquisition of a mathematical reasoning score of 35.81 (poor), while the easiest mastered by students is "Presenting mathematical statements or premises in writing, drawings or graphics" with mathematical reasoning score of 80.14 (Very Good). The students' mathematical reasoning ability on the indicator "compiling an analogy or explanation of the problem and its solution" is 74.09 (Good). The mathematical reasoning ability of students on the indicator "mathematical manipulation" is 42.10 (fair).&#x0D; Keywords: Mathematical Reasoning, Students’ ability, Class Wide Peer Tutoring

Robust recursive absolute value inequalities discriminant analysis with sparseness

In this paper, we propose a novel absolute value inequalities discriminant analysis (AVIDA) criterion for supervised dimensionality reduction. Compared with the conventional linear discriminant analysis (LDA), the main characteristics of our AVIDA are robustness and sparseness. By reformulating the generalized eigenvalue problem in LDA to a related SVM-type “concave-convex” problem based on absolute value inequalities loss, our AVIDA is not only more robust to outliers and noises, but also avoids the SSS problem. Moreover, the additional L1-norm regularization term in the objective makes sure sparse discriminant vectors are obtained. A successive linear algorithm is employed to solve the proposed optimization problem, where a series of linear programs are solved. The superiority of our AVIDA is supported by experimental results on artificial examples as well as benchmark image databases.

Unsupervised and Semisupervised Classification Via Absolute Value Inequalities

We consider the problem of classifying completely or partially unlabeled data by using inequalities that contain absolute values of the data. This allows each data point to belong to either one of two classes by entering the inequality with a plus or minus value. By using such absolute value inequalities in linear and nonlinear support vector machines, unlabeled or partially labeled data can be successfully partitioned into two classes that capture most of the correct labels dropped from the unlabeled data.

Unsupervised classification via convex absolute value inequalities

We consider the problem of classifying completely unlabelled data using convex inequalities that contain absolute values of the data. This allows each data point to belong to either one of two classes by entering the inequality with a plus or minus value. Using such absolute value inequalities in support vector machine classifiers, unlabelled data can be successfully partitioned into two classes that capture most of the correct labels dropped from the data. Inclusion of partially labelled data leads to a semisupervised classifier. Computational results include unsupervised and semisupervised classification of the Wisconsin Breast Cancer Wisconsin (Diagnostic) Data Set.

Algebraic Activities Aid Discovery Lessons

Three graphing activities lead students to discover the shapes and properties of the graphs for linear, quadratic, and absolute value functions and inequalities.

Counterexamples as Starting Points for Reasoning and Sense Making

Asked to “fix” a false conjecture, students combine their reasoning and observations about absolute value inequalities, signed numbers, and distance to write true mathematical statements.

Absolute value inequalities: high school students’ solutions and misconceptions

Inequalities are one of the foundational subjects in high school math curricula, but there is a lack of academic research into how students learn certain types of inequalities. This article fills part of the research gap by presenting the findings of a study that examined high school students’ methods of approaching absolute value inequalities, their common mistakes, misconceptions, and the possible sources of these mistakes and misconceptions. The research study used two tools—a questionnaire and personal interviews. The questionnaire was given to 481 students in the 10th and 11th grades in Israel who studied mathematics at intermediate and advanced levels. It was administered after the students had studied inequalities. Thirty-two students were interviewed in order to find their ways of thinking and the sources of their errors. The main types of mistakes that students consistently made when solving absolute value inequalities were found. Based on the study’s findings, teachers can understand students’ thought processes and use this understanding to conduct remediation and enhance mathematics instruction.

A short note on solvability of systems of interval linear equations

In this note we formulate necessary and sufficient conditions for strong solvability and feasibility of systems of linear interval equations in terms of absolute value inequalities.

Teaching absolute value inequalities to mature students

This paper gives an account of a teaching experiment on absolute value inequalities, whose aim was to identify characteristics of an approach that would realize the potential of the topic to develop theoretical thinking in students enrolled in prerequisite mathematics courses at a large, urban North American university. The potential is demonstrated in an epistemological analysis of the topic. It is also shown that this potential is not realized in the way the topic is presently taught in prerequisite mathematics courses. Three groups of students enrolled in such courses were each exposed to one of three approaches we conceived for teaching the topic, labeled the procedural (PA), the theoretical (TA), and the visual (VA) approaches. The design of the three lectures was constrained by institutional characteristics of college-level courses, and informed by epistemological and didactical analyses of the topic. It was found that following the VA lecture, which proposed two equally valid mathematical techniques (graphical and analytic), one of which could be used to test the validity of results obtained by the other, students were more likely to engage in some aspects of theoretical thinking. They displayed more reflective and systemic thinking than other groups, and dealt more effectively with the logical intricacies of absolute value inequalities. VA students appeared to have a synthetic grasp of the inequalities, and a flexibility of thought not displayed by PA and TA students. However, without sufficient attention to tasks not easily solved by graphical means, VA approach provided students with a way to avoid the challenges of systemic and analytic thinking, some of which were more apparent in TA students. PA students expectedly behaved more as procedural knowers, but we saw interesting examples of engagement with theoretical thinking while dealing with the procedures proposed in the PA lecture.

Forty necessary and sufficient conditions for regularity of interval matrices: A survey

This is a survey of forty necessary and sufficient conditions for regularity of interval matrices published in various papers over the last thirty-five years. Afull list of references to the sources of all the conditions is given, and they are commented on in detail. 1. Introduction. During the last thirty-five years (1973-2008), considerable in- terest has been dedicated to the problemof regularity of interval m atrices. It has resulted in formulations of altogether forty necessary and sufficient conditions that constitute the subject matter of this survey paper. By definition, a square interval matrix A is called regular if each A ∈ A is nonsingular, and it is said to be singular otherwise (i.e., if it contains a singular matrix). It is the purpose of this paper to show that this property can be reformulated in surprisingly many surprisingly various ways. In the main Theorem 4.1 we show that regularity of interval matrices can be characterized in terms of determinants (Theorem 4.1, condition (xxxii)), matrix inverses (xxx), linear equations (xxv), absolute value equations (v), absolute value inequalities (ii), matrix equations (xxiv), solvability in each orthant (xvi), inclusions (xxxvii), set properties (xxxvi), real spectral radius (xxxiv), P -matrices (xxix) and edge nonsingularity (xli). We do not include the proof of mutual equivalence of all the conditions since that would make for a very lengthy

Absolute value programming

We investigate equations, inequalities and mathematical programs involving absolute values of variables such as the equation Ax+B|x| = b, where A and B are arbitrary m× n real matrices. We show that this absolute value equation is NP-hard to solve, and that solving it with B = I solves the general linear complementarity problem. We give sufficient optimality conditions and duality results for absolute value programs as well as theorems of the alternative for absolute value inequalities. We also propose concave minimization formulations for absolute value equations that are solved by a finite succession of linear programs. These algorithms terminate at a local minimum that solves the absolute value equation in almost all solvable random problems tried.

Using MathCAD to teach one-dimensional graphs

Topics such as linear and nonlinear equations and inequalities, compound inequalities, linear and nonlinear absolute value equations and inequalities, rational equations and inequality are commonly found in college algebra and precalculus textbooks. What is common about these topics is the fact that their solutions and graphs lie in the real line one-dimensional. However, sketching these graphs using computer software and graphing utilities is not straightforward. In this note we show how to use MathCAD to address this problem. The approach is simple and can be used by teachers in teaching almost all topics whose graphs and solutions lie in the real line. The method encourages students to explore mathematical models in these topics.

The interaction of spatial visualization and general reasoning abilities with instructional treatment in absolute value inequalities

This study is one in a series that have used the aptitude variable of spatial visualization and general reasoning abilities. One hundred and six college students were randomly assigned to two treatments on the topic of absolute value inequalities. The algebraic treatment was designed to use the algebraic definition of absolute value and the goemetric treatment was designed to use the geometric definition of absolute value as a distance on the number line. It was hypothesized that subjects having good general reasoning ability would do better in the algebraic treatment and subjects having good spatial visualization ability would do better in the geometric treatment. The analysis was done by multiple regression technique. No interactions were found between the dependent variable of achievement on absolute value equations and inequalities and the aptitude variable. However, general reasoning was a significant predictor for the algebraic treatment group on the equation test and spatial visualization was a sig...

The Graphing of Absolute Value Inequalities

When dealing with absolute value inequalities one is confronted with the problem of solving a pair of inequalities. For some pairs the solution results in the union of two sets, while for other pairs the solution results in the intersection of two sets.

The logic of absolute value inequalities

Teachers motivated by the need to express the distance of any point on the real line from zero find the definition of absolute value useful and interesting. It is used in elementary algebra to define addition and multiplication of “directed” numbers and to describe the graphs of broken lines in one and two dimensions. Recent emphasis on inequalities and the inclusion in newer textbooks of absolute value inequalities raise several questions that cannot be handled in a perfunctory manner.