Name: absolute value inequalities on a number line
Uploaded: 2025-11-26T05:44:25.000Z
Duration: 1 min 23 s
Channel: JoVE

Der absolute Wert gibt den Abstand an, in dem eine Zahl auf dem Zahlenstrahl von Null entfernt ist, unabhängig davon, ob die Zahl positiv oder negativ ist.Wenn der absolute Wert einer Größe kleiner oder gleich einem bestimmten Wert ist, schließt die Lösung alle Zahlen zwischen der negativen und der positiven Grenze ein.Wenn der absolute Wert größer oder gleich einem bestimmten Wert ist, schließt die Lösung alle Zahlen außerhalb dieser Grenzen ein.Diese Lösungen werden auf einem Zahlenstrahl mit Segmenten oder Strahlen dargestellt, wobei Kreise Endpunkte anzeigen, die aufgrund des Gleichheitsteils der Ungleichung enthalten sind.Erwägen Sie, die Schallgeschwindigkeit mithilfe von Echos aus einer bekannten Entfernung zu einer Wand zu messen. Abhängig von Schwankungen der Umgebung liegt die gemessene Geschwindigkeit bei etwa 343 Metern pro Sekunde, mit einer Variation von plus/minus 2 Metern pro Sekunde.Diese Variation wird anhand einer absoluten Wertungleichung modelliert, die eine Grenze dafür festlegt, wie weit die beobachtete Geschwindigkeit vom erwarteten Wert abweichen kann.Um die Ungleichung zu lösen, wird die zulässige Variation subtrahiert und zur erwarteten Geschwindigkeit addiert, um den gesamten Bereich der akzeptablen Werte zu erhalten.

betragsungleichungen

Der Betrag ist ein mathematisches Ma&#223;, das den Abstand einer Zahl von Null auf der Zahlengeraden unabh&#228;ngig von ihrem Vorzeichen angibt. Im Kontext von Ungleichungen helfen Betragsausdr&#252;cke dabei, einen Bereich zul&#228;ssiger Werte beziehungsweise Grenzen f&#252;r eine Variable zu bestimmen. Diese Ungleichungen werden h&#228;ufig in der wissenschaftlichen Modellierung und Dateninterpretation eingesetzt, wo Variabilit&#228;t innerhalb oder jenseits eines bestimmten Schwellenwerts pr&#228;zise erfasst werden muss.Eine Betragsungleichung der Form &#8739;x&#8739; &#8804; a, wobei a &#8805; 0, beschreibt alle Werte von x zwischen &#8722;a und a einschlie&#223;lich der Endpunkte. Diese Ungleichung l&#228;sst sich als zusammengesetzte Ungleichung schreiben:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:52.41%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40036/40036_Equation_1_2__editor_40036.svg' alt='40036_Equation_1_2_.svg' /></figure>Grafisch wird dieser Bereich auf der Zahlengeraden durch eine durchgezogene Strecke dargestellt, die die Endpunkte bei &#8722;a und a verbindet; h&#228;ufig sind die Endpunkte mit geschlossenen Kreisen markiert, um ihre Zugeh&#246;rigkeit zur L&#246;sung zu kennzeichnen.Umgekehrt bedeutet eine Ungleichung der Form &#8739;x&#8739; &#8805; a, dass x einen Abstand von mindestens a zu Null hat. Diese L&#246;sungsmenge umfasst zwei disjunkte Intervalle:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:53.34%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40036/40036_Equation_2_2__editor_40036.svg' alt='40036_Equation_2_2_.svg' /></figure>Diese werden typischerweise durch Halbgeraden dargestellt, die von den Endpunkten nach au&#223;en verlaufen, h&#228;ufig mit geschlossenen Kreisen markiert, wenn die Endpunkte eingeschlossen sind.Betrachten Sie eine chemische Reaktion, die bei 80 &#176;C optimal abl&#228;uft. Um Sicherheit und Ausbeute zu gew&#228;hrleisten, muss die Temperatur innerhalb von 2,5 &#176;C um diesen Wert liegen. Diese Anforderung l&#228;sst sich wie folgt ausdr&#252;cken:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40036/40036_Equation_3_2__editor_40036.svg' alt='40036_Equation_3_2_.svg' /></figure>Diese Ungleichung bedeutet, dass die Temperatur x h&#246;chstens um 2,5 &#176;C vom Idealwert abweichen darf. Die L&#246;sung der Ungleichung lautet:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40036/40036_Equation_4_2__editor_40036.svg' alt='40036_Equation_4_2_.svg' /></figure>Daher gew&#228;hrleistet jede Temperatur innerhalb dieses Intervalls optimale Reaktionsbedingungen.

Betragsungleichungen

Der Betrag ist ein mathematisches Ma&#223;, das den Abstand einer Zahl von Null auf der Zahlengeraden unabh&#228;ngig von ihrem Vorzeichen angibt. Im ...

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Absolute Value Inequalities

absolute value inequalities on a number line

Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable.

An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at the endpoints, because the endpoints are part of the answer.

An inequality of the form |x| ≥ a means x is at least a units away from zero. Its solution has two separate intervals, one on each side of zero. These are usually drawn as rays extending outward from the endpoints, and closed circles are used when the endpoints are included.

These inequalities are useful in scientific modeling and data interpretation, where values must stay within a set limit or beyond a boundary. A chemical reaction that works best at 80°C can be used as an example. If the temperature must stay within 2.5°C of 80°C, the condition can be written as an absolute value inequality.

That inequality shows that the temperature x may vary only a small amount from the ideal value. Solving it gives an interval of acceptable temperatures. Any temperature in that interval keeps the reaction in its optimal range.

Absolute Value Inequalities on a Number Line

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Learn how absolute value inequalities show distance from zero, form compound inequalities, and appear on a number line with a science-based temperature example.

Absolute Value Inequalities on a Number Line | JoVE K12

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Absolute Value Inequalities on a Number Line

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