Name: graphing and solving nonlinear inequalities
Uploaded: 2025-11-26T05:43:49.000Z
Duration: 1 min 25 s
Channel: JoVE

Dans l’inégalité linéaire, la frontière décrit une ligne droite qui sépare la région de solution de la région qui satisfait l’autre condition de l’inégalité.En revanche, une inégalité non linéaire implique des expressions de degré supérieur et définit des régions à l’aide de frontières courbes au lieu de lignes droites.Lorsque l’on travaille avec de telles inégalités, en particulier les types polynomiaux, des solutions sont souvent trouvées en factorisant l’expression et en fixant chaque facteur égal à zéro pour déterminer les racines, ou zéros.Ces points ne sont pas inclus dans la solution mais indiquent où l’expression change de signe, divisant la droite numérique en intervalles dont la validité peut être testée dans l’inéquation.Une valeur de chaque intervalle est substituée dans l’inégalité d’origine pour déterminer si l’expression est vraie dans cette région.Les intervalles où l’inégalité est satisfaite définissent les régions de solution.Dans un contexte commercial, le bénéfice est la différence entre le chiffre d’affaires et le coût des unités vendues. Pour assurer un bénéfice d’au moins 20 dollars, la condition est exprimée sous la forme d’une inégalité non linéaire. La résolution de cette inégalité donne la plage spécifique d’unités pour lesquelles l’exigence de profit est satisfaite.

introduction aux inégalités non linéaires

Les in&#233;galit&#233;s lin&#233;aires et non lin&#233;aires sont fondamentales pour analyser les relations entre variables et identifier les intervalles satisfaisant des conditions sp&#233;cifiques. Une in&#233;galit&#233; lin&#233;aire porte sur des variables au premier degr&#233;, ce qui se traduit par le graphe d&#8217;une droite. Cette droite partage le plan cart&#233;sien en deux r&#233;gions distinctes&#160;: l&#8217;une qui satisfait l&#8217;in&#233;galit&#233; et l&#8217;autre qui ne la satisfait pas. Chaque r&#233;gion correspond &#224; un ensemble de solutions pour lequel la relation lin&#233;aire demeure vraie sous la contrainte sp&#233;cifi&#233;e.
Les in&#233;galit&#233;s non lin&#233;aires s&#8217;en distinguent en faisant intervenir des puissances strictement sup&#233;rieures &#224; un. Elles incluent les in&#233;galit&#233;s quadratiques et, plus g&#233;n&#233;ralement, polynomiales de degr&#233; sup&#233;rieur, qui produisent des courbes. Ces courbes traduisent des relations o&#249; le taux de variation n&#8217;est pas constant, mais d&#233;pend de la variable. La r&#233;solution de ces in&#233;galit&#233;s consiste &#224; examiner le comportement de la fonction sur diff&#233;rents intervalles de la droite r&#233;elle.
Les racines &#8212; valeurs pour lesquelles l&#8217;expression est &#233;gale &#224; z&#233;ro &#8212; marquent les endroits o&#249; la fonction peut changer de signe. Ces points d&#233;coupent la droite r&#233;elle en intervalles, dont chacun peut &#234;tre test&#233; en substituant une valeur repr&#233;sentative dans l&#8217;in&#233;galit&#233; initiale. Les intervalles pour lesquels l&#8217;in&#233;galit&#233; est v&#233;rifi&#233;e constituent l&#8217;ensemble des solutions.
Dans des contextes appliqu&#233;s tels que l&#8217;&#233;conomie ou la gestion, les in&#233;galit&#233;s non lin&#233;aires offrent des &#233;clairages pr&#233;cieux. Par exemple, une fonction de profit influenc&#233;e par le volume de production et les co&#251;ts marginaux peut &#234;tre mod&#233;lis&#233;e par une expression non lin&#233;aire. En appliquant l&#8217;in&#233;galit&#233;, on identifie l&#8217;intervalle de ventes garantissant que le profit demeure sup&#233;rieur &#224; un seuil donn&#233;, par exemple 500 $, orientant ainsi efficacement la prise de d&#233;cision sur la base d&#8217;une analyse math&#233;matique.

Introduction aux inégalités non linéaires

Les in&#233;galit&#233;s lin&#233;aires et non lin&#233;aires sont fondamentales pour analyser les relations entre variables et identifier les intervalles ...

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Introduction to Nonlinear Inequalities

graphing and solving nonlinear inequalities

Linear and nonlinear inequalities help describe variable relationships and the ranges that satisfy a condition. A linear inequality uses variables raised only to the first power. Its graph is a straight line that divides the coordinate plane into two regions. One region satisfies the inequality, and the other does not.

Nonlinear inequalities use variables raised to powers greater than one. These include quadratic inequalities and higher-degree polynomial inequalities. Their graphs are curved, not straight. The changing shape shows that the rate of change is not constant as the input changes.

Solving a nonlinear inequality often means looking at intervals on the number line. Critical points are the values where the expression equals zero. These points show where the function may change sign. They split the number line into sections that can be tested with a representative value from each interval.

The intervals that make the inequality true form the solution set. This approach is useful in applied settings such as economics and business. A profit function can depend on production volume and marginal costs. By using a nonlinear inequality, it is possible to find the sales range that keeps profit above a target such as five hundred dollars.

Graphing and Solving Nonlinear Inequalities

graphing-and-solving-nonlinear-inequalities

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introduction to nonlinear inequalities

Linear and nonlinear inequalities help describe variable relationships and the ranges that satisfy a condition. A linear inequality uses variables raised only to the first power. Its graph is a straight line that divides the coordinate plane into two regions. One region satisfies the inequality, and the other does not. Nonlinear inequalities use variables raised to powers greater than one. These include quadratic inequalities and higher-degree polynomial inequalities. Their graphs are curved,...

application of nonlinear inequalities

Solving Nonlinear Inequalities on a Number Line

Nonlinear inequalities can be solved by checking where an expression is positive or negative on a number line. These inequalities often include squares, products, or variables in the denominator, so they do not follow the same simple pattern as linear inequalities. The first step is to rewrite the inequality so that zero is on one side. After rewriting, factor the expression if possible. Factoring helps you find the key values where the expression equals zero. In one example, the values 3 and...

key terms and concepts

Learn how linear and nonlinear inequalities differ, how critical points split a number line, and how interval testing finds the solution set.

Graphing and Solving Nonlinear Inequalities | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Graphing and Solving Nonlinear Inequalities

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