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

Bei einer vertikalen Projektilbewegung ohne Luftwiderstand wird ein Objekt mit einer Geschwindigkeit von 50 Metern pro Sekunde vom Boden aus abgeschossen. Seine Höhe über die Zeit folgt der quadratischen Gleichung: minus fünf t zum Quadrat plus fünfzig t, wobei minus fünf die Hälfte der Erdbeschleunigung ist. Ziel ist es, das Zeitintervall zu bestimmen, in dem die Höhe des Projektils 100 Meter überschreitet.Da die Höhenfunktion quadratisch ist, muss beim Vergleich mit einem konstanten Wert eine quadratische Ungleichung gelöst werden.Die Ungleichung wird neu angeordnet, indem alle Terme auf eine Seite verschoben und durch minus fünf dividiert werden, wodurch der führende Koeffizient auf eins vereinfacht wird.Da der resultierende Ausdruck nicht einfach faktorisiert werden kann, wird die quadratische Formel verwendet. Die Koeffizienten werden in die Formel eingesetzt, um zwei Lösungspunkte zu finden.Diese Werte stellen den Anfang und das Ende des Zeitraums dar, in dem sich das Projektil über 100 Meter aufhält.Ein Diagramm der Höhenfunktion hilft dabei, diese Situation zu visualisieren. Die Kurve steigt über die 100-Meter-Marke, erreicht einen Gipfel und fällt dann wieder auf die Marke zurück.Im Diagramm wird der Teil der Kurve, der über 100 Metern liegt, schattiert, um das Zeitintervall anzuzeigen, in dem die Bedingung erfüllt ist.

nichtlineare ungleichungen

Eine nichtlineare Ungleichung beschreibt eine Ungleichung mit einem nichtlinearen Ausdruck, der gekr&#252;mmt ist oder sich komplexer verh&#228;lt als eine Gerade. Diese Ungleichungen treten h&#228;ufig in Formen auf, die quadratische Terme, Produkte oder Variablen im Nenner enthalten.Zur L&#246;sung schreibt man die Ungleichung zun&#228;chst so um, dass auf einer Seite die Null steht. Beispiel:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><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>kann wie folgt faktorisiert werden:<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>Diese Form erleichtert die Identifizierung der Werte, f&#252;r die der Ausdruck Null wird. In diesem Fall sind die Nullstellen 3 und &#8722;2. Diese Werte unterteilen die Zahlengerade in drei Bereiche.W&#228;hlen Sie aus jedem Bereich einen Probewert und setzen Sie ihn in den Ausdruck ein. Dieser Vorzeichentest zeigt, ob der Ausdruck in dem jeweiligen Abschnitt positiv oder negativ ist. Verlangt die urspr&#252;ngliche Ungleichung, dass der Ausdruck kleiner als Null ist, dann sind die richtigen Bereiche diejenigen mit negativem Ergebnis. Die vollst&#228;ndige L&#246;sung umfasst nur diese Bereiche.Derselbe Ansatz gilt f&#252;r rationale Ungleichungen, bei denen die Variable im Nenner steht. Dabei ist zus&#228;tzlich zu ber&#252;cksichtigen, wo der Ausdruck nicht definiert ist. Beispiel:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:98.75%;'><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>In diesem Fall sind die kritischen Stellen &#8722;1 und 4. Diese unterteilen die Zahlengerade in disjunkte Intervalle. Testen Sie jedes Intervall und schlie&#223;en Sie den Punkt aus, an dem der Nenner Null wird &#8211; hier also 4.Nichtlineare Ungleichungen sind in der Praxis n&#252;tzlich. Eine Wirtschaftsanalystin oder ein Wirtschaftsanalyst k&#246;nnte damit bestimmen, wann der Gewinn &#252;ber Null bleibt. Eine Ingenieurin oder ein Ingenieur k&#246;nnte damit sicherstellen, dass eine Maschine innerhalb sicherer Druckgrenzen arbeitet. Eine Physikerin oder ein Physiker k&#246;nnte damit analysieren, wann ein Objekt innerhalb eines bestimmten Bewegungsbereichs bleibt.

Nichtlineare Ungleichungen

Eine nichtlineare Ungleichung beschreibt eine Ungleichung mit einem nichtlinearen Ausdruck, der gekr&#252;mmt ist oder sich komplexer verh&#228;lt als ...

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Polynomial and Rational Functions

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KERNBEGRIFFE UND KONZEPTE

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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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

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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