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

El valor absoluto indica la distancia a la que se encuentra un número de cero en la recta numérica, independientemente de si el número es positivo o negativo.Cuando el valor absoluto de una cantidad es menor o igual que un valor dado, la solución incluye todos los números entre los límites negativo y positivo.Si el valor absoluto es mayor o igual que un valor dado, la solución incluye todos los números fuera de esos límites.Estas soluciones se muestran en una recta numérica usando segmentos o rayos, con círculos que indican los puntos finales incluidos debido a la parte de igualdad de la desigualdad.Considere medir la velocidad del sonido usando ecos desde una distancia conocida a una pared. Dependiendo de las fluctuaciones ambientales, la velocidad medida es de alrededor de 343 metros por segundo, con una variación de más o menos 2 metros por segundo.Esta variación se modela utilizando una desigualdad de valor absoluto, que establece un límite sobre cuánto puede desviarse la velocidad observada del valor esperado.Resolver la desigualdad implica restar y sumar la variación permitida a la velocidad esperada para obtener el rango completo de valores aceptables.

desigualdades de valor absoluto

El valor absoluto es un concepto matem&#225;tico que representa la distancia de un n&#250;mero al cero en la recta num&#233;rica, independientemente de su signo. En el contexto de las desigualdades, las expresiones con valor absoluto ayudan a definir un rango de valores permitidos o l&#237;mites para una variable. Estas desigualdades se utilizan com&#250;nmente en el modelado cient&#237;fico y la interpretaci&#243;n de datos, donde la variabilidad dentro o fuera de un umbral determinado debe representarse con precisi&#243;n.Una desigualdad de valor absoluto de la forma &#8739;x&#8739; &#8804; a, donde a &#8805; 0, describe todos los valores de x entre &#8722;a y a, ambos incluidos. Esta desigualdad puede reescribirse como una desigualdad compuesta:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:40.45%;'><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>Gr&#225;ficamente, este rango se representa en una recta num&#233;rica con un segmento de l&#237;nea continua que conecta los extremos en &#8722;a y a, a menudo marcados con c&#237;rculos cerrados para indicar que los extremos est&#225;n incluidos en la soluci&#243;n.Por el contrario, una desigualdad de la forma &#8739;x&#8739; &#8805; a indica que x se encuentra a una distancia de al menos una unidad del cero. Este conjunto de soluciones incluye dos intervalos separados:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:40.8%;'><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>Estos se representan t&#237;picamente mediante rayos que se extienden hacia el exterior desde los extremos, a menudo marcados con c&#237;rculos cerrados si los extremos est&#225;n incluidos.Considere una reacci&#243;n qu&#237;mica que procede de forma &#243;ptima a 80 &#176;C. Para garantizar la seguridad y la eficiencia del proceso, la temperatura debe mantenerse dentro de 2,5 &#176;C de ese valor. Este requisito puede expresarse como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:40.1%;'><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>Esta desigualdad significa que la temperatura x puede desviarse como m&#225;ximo 2,5 &#176;C del valor ideal. Al resolver la desigualdad se obtiene:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:40.45%;'><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>Por lo tanto, cualquier temperatura dentro de este intervalo mantiene las condiciones &#243;ptimas de reacci&#243;n.

Desigualdades de valor absoluto

El valor absoluto es un concepto matem&#225;tico que representa la distancia de un n&#250;mero al cero en la recta num&#233;rica, independientemente de su ...

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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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absolute value inequalities

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key terms and concepts

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

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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Absolute Value Inequalities on a Number Line

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