Name: solving nonlinear inequalities on a number line
Uploaded: 2025-11-26T05:43:48.000Z
Duration: 1 min 29 s
Channel: JoVE

Dans un mouvement de projectile vertical sans résistance de l’air, un objet est lancé depuis le sol à une vitesse de 50 mètres par seconde. Sa hauteur au fil du temps suit l’équation du second degré : moins cinq t au carré plus cinquante t, où moins cinq est la moitié de l’accélération due à la gravité. L’objectif est de déterminer l’intervalle de temps lorsque la hauteur du projectile dépasse 100 mètres.Comme la fonction de hauteur est quadratique, la comparer à une valeur constante implique de résoudre une inégalité quadratique.L’inégalité est réorganisée en déplaçant tous les termes d’un côté et en divisant par moins cinq, ce qui simplifie le coefficient principal à un.Comme l’expression résultante ne peut pas être factorisée facilement, la formule du second degré est utilisée. Les coefficients sont substitués dans la formule pour trouver deux points de solution.Ces valeurs représentent le début et la fin de la période pendant laquelle le projectile reste au-dessus de 100 mètres.Un graphique de la fonction height permet de visualiser cette situation. La courbe s’élève au-dessus de la marque des 100 mètres, atteint un sommet, puis redescend jusqu’à la marque.Sur le graphique, la partie de la courbe qui se trouve au-dessus de 100 mètres est ombrée pour montrer l’intervalle de temps pendant lequel la condition est remplie.

inégalités non linéaires

Une in&#233;quation non lin&#233;aire d&#233;crit une comparaison mettant en jeu une expression non affine, au comportement plus complexe qu&#8217;une droite. Ces in&#233;quations se pr&#233;sentent souvent sous des formes incluant des carr&#233;s, des produits ou des variables au d&#233;nominateur.Pour r&#233;soudre une telle in&#233;quation, on commence par la r&#233;&#233;crire de fa&#231;on &#224; faire appara&#238;tre 0 d&#8217;un c&#244;t&#233;. Par exemple, l&#8217;in&#233;quation suivante&#160;:<figure class='image image-style-align-center' style='display:table;margin-left:auto;margin-right:auto;'><img style='aspect-ratio:1920/74;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_1_editor_40060.svg' alt='40060_Equation_1.svg' /></figure>peut &#234;tre factoris&#233;e comme suit&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:100%;'><img style='aspect-ratio:300/9;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_2_editor_40060.svg' alt='40060_Equation_2.svg' /></figure>Cette &#233;criture facilite l&#8217;identification des valeurs qui annulent l&#8217;expression. Dans ce cas, les valeurs critiques sont 3 et &#8722;2. Ces valeurs divisent la droite r&#233;elle en trois intervalles.On choisit une valeur d&#8217;essai dans chaque intervalle et on &#233;value l&#8217;expression en cette valeur. Ce test indique si l&#8217;expression est positive ou n&#233;gative sur cet intervalle. Si l&#8217;in&#233;quation initiale impose que l&#8217;expression soit strictement inf&#233;rieure &#224; 0, les intervalles corrects sont ceux o&#249; le r&#233;sultat est n&#233;gatif. La solution compl&#232;te est alors l&#8217;union de ces intervalles.La m&#234;me d&#233;marche s&#8217;applique aux in&#233;quations rationnelles, o&#249; la variable appara&#238;t au d&#233;nominateur. Cependant, il faut aussi tenir compte des points o&#249; l&#8217;expression n&#8217;est pas d&#233;finie. Par exemple&#160;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:100%;'><img style='aspect-ratio:300/20;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_3_editor_40060.svg' alt='40060_Equation_3.svg' /></figure>Dans ce cas, les valeurs critiques sont &#8722;1 et 4. Elles divisent la droite r&#233;elle en intervalles distincts. On teste chaque intervalle et l&#8217;on exclut le point o&#249; le d&#233;nominateur s&#8217;annule &#8212; ici, 4.Les in&#233;quations non lin&#233;aires sont utiles en pratique. Un analyste d&#8217;entreprise peut en utiliser une pour d&#233;terminer quand le b&#233;n&#233;fice reste strictement positif. Un ing&#233;nieur peut y recourir pour s&#8217;assurer qu&#8217;une machine fonctionne dans des plages de pression s&#251;res. Un physicien peut analyser &#224; quel moment un objet demeure dans un certain intervalle de mouvement.

Inégalités non linéaires

Une in&#233;quation non lin&#233;aire d&#233;crit une comparaison mettant en jeu une expression non affine, au comportement plus complexe qu&#8217;une ...

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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 -2 split the number line into three regions. A sample value from each region is then tested in the expression to see whether that interval gives a positive or negative result.

The final answer includes only the regions that match the original sign of the inequality. If the inequality says the expression must be less than zero, then only the intervals with negative results are included. The same interval test works for rational inequalities, where the variable appears in the denominator. These problems also require checking where the expression is undefined, so any value that makes the denominator zero must be left out.

For example, a rational inequality may have key values of -1 and 4. These values divide the number line into separate intervals. Each interval is tested, but the point where the denominator becomes zero, such as 4, is excluded from the solution. This process is useful in real situations, such as finding when profit stays above zero, keeping machine pressure within safe limits, or checking when an object stays within a certain range of motion.

Solving Nonlinear Inequalities on a Number Line

solving-nonlinear-inequalities-on-a-number-line

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application of nonlinear inequalities

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 to solve nonlinear and rational inequalities by factoring, testing number line intervals, and excluding undefined values in grade 9–12 math.

Solving Nonlinear Inequalities on a Number Line | 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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Solving Nonlinear Inequalities on a Number Line

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