Name: solving inequalities and interval notation
Uploaded: 2025-11-26T05:44:26.000Z
Duration: 1 min 28 s
Channel: JoVE

Bei Ungleichungen werden zwei Werte verglichen, um ihre Beziehung mithilfe von Symbolen wie "Kleiner als", "Größer als", "Kleiner als" oder "Gleich" und "Größer als" oder "Gleich" anzuzeigen.Intervallnotationen stellen den Wertebereich dar, der eine Ungleichung als prägnante Form von Lösungsmengen erfüllt.Ein geschlossenes Intervall umfasst die Endpunkte und wird in eckigen Klammern angezeigt, während ein offenes Intervall die Endpunkte nicht enthält und in Klammern angezeigt wird.Zu den Eigenschaften von Ungleichungen gehört die Regel, dass durch das Addieren oder Subtrahieren desselben Wertes auf beiden Seiten die Richtung der Ungleichung gleich bleibt.Das Multiplizieren beider Seiten mit einer positiven Zahl oder das Dividieren beider Seiten durch die gleiche Zahl behält ebenfalls die Richtung bei, aber das Multiplizieren oder Dividieren durch eine negative Zahl kehrt die Ungleichung um.Wenn beide Seiten einer Ungleichung positiv sind, kehrt sich die Richtung der Ungleichheit um, wenn man Kehrwerte nimmt. Diese Regel gilt nicht, wenn die Werte negativ sind.Die Vereinfachung einer Ungleichung besteht darin, die Konstante zu entfernen und dann beide Seiten durch eine Konstante zu dividieren, um die Variable zu isolieren.Diese Ungleichheiten treten im wirklichen Leben auf. Die Regel "Über 100 cm hoch" in einem Fahrgeschäft lässt beispielsweise nur Personen zu, die größer als 100 cm sind.

ungleichungen

Ungleichungen dr&#252;cken mathematische Beziehungen aus, bei denen zwei Werte ungleich sind und mit Symbolen wie &lt;, &gt;, &#8804; oder &#8805; verglichen werden. Diese Ausdr&#252;cke definieren einen Bereich m&#246;glicher L&#246;sungen anstelle eines einzelnen Werts. Die Intervallnotation bietet eine knappe M&#246;glichkeit, diese L&#246;sungsmengen auszudr&#252;cken, insbesondere wenn die Variable einen kontinuierlichen Bereich umfasst. Ein offenes Intervall, geschrieben als (a, b), schlie&#223;t die Endpunkte aus, w&#228;hrend ein geschlossenes Intervall [a, b] sie einschlie&#223;t. Es gibt auch halboffene Intervalle wie (a, b] und [a, b), die nur einen Endpunkt enthalten. Die Notationen (a, &#8734;) und (&#8722;&#8734;, b] werden verwendet, wenn sich eine Menge in eine Richtung unendlich erstreckt, wobei die Unendlichkeitssymbole niemals als tats&#228;chliche Werte enthalten sind.
Das L&#246;sen von Ungleichungen erfordert die Anwendung von Operationen unter Einhaltung bestimmter Regeln, die die Richtung der Ungleichung beibehalten oder umkehren. Wenn eine Konstante zu beiden Seiten addiert oder von beiden Seiten subtrahiert wird, bleibt die Richtung der Ungleichung unver&#228;ndert. Dies wird durch die Additionsregel formalisiert: Gilt a &lt; b, dann gilt auch a + c &lt; b + c. Ebenso beh&#228;lt die Multiplikation beider Seiten mit einer positiven Zahl bzw. die Division durch eine positive Zahl die Richtung der Ungleichung bei, w&#228;hrend sie bei Verwendung einer negativen Zahl umgekehrt wird. Beispielsweise ergibt die Multiplikation beider Seiten von 3 &lt; 7 mit 2 die Ungleichung 6 &lt; 14, w&#228;hrend die Multiplikation beider Seiten von 3 &lt; 7 mit &#8722;2 die Ungleichung &#8722;6 &gt; &#8722;14 ergibt.
Die Reziprokeigenschaft von Ungleichungen spielt ebenfalls eine entscheidende Rolle. Gilt a &gt; 0, dann gilt auch 1/a &gt; 0; bildet man bei einer Ungleichung mit positiven Zahlen beiderseits den Kehrwert, kehrt sich die Richtung der Ungleichung um. Sind beispielsweise a &lt; b und beide positiv, dann ist 1/a &gt; 1/b.

Ungleichungen

Ungleichungen dr&#252;cken mathematische Beziehungen aus, bei denen zwei Werte ungleich sind und mit Symbolen wie &lt;, &gt;, &#8804; oder &#8805; ...

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KEY TERMS AND CONCEPTS

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Inequalities

solving inequalities and interval notation

Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value.

Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint.

Intervals can also extend forever in one direction. The forms (a, ∞) and (−∞, b] are used for these cases. The infinity symbols do not count as actual values.

Solving inequalities depends on keeping the inequality sign or reversing it at the right time. If the same constant is added or subtracted from both sides, the direction stays the same. This is the addition property: if a −14 when both sides are multiplied by −2.

The reciprocal property is also important for positive values. If a &gt; 0, then 1/a &gt; 0. For positive numbers, taking reciprocals reverses the inequality direction. So if a 1/b.

Solving Inequalities and Interval Notation

solving-inequalities-and-interval-notation

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Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value. Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint. Intervals can also extend forever in one...

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key terms and concepts

Learn how inequalities, interval notation, and inequality rules work. See how open, closed, and half-open intervals describe solution sets.

Solving Inequalities and Interval Notation | 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.

math fundamentals

Solving Inequalities and Interval Notation

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