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

En un movimiento de proyectil vertical sin resistencia del aire, se lanza un objeto desde el nivel del suelo con una velocidad de 50 metros por segundo. Su altura a lo largo del tiempo sigue la ecuación cuadrática: menos cinco t al cuadrado más cincuenta t, donde menos cinco es la mitad de la aceleración debida a la gravedad. El objetivo es determinar el intervalo de tiempo en el que la altura del proyectil supera los 100 metros.Dado que la función de altura es cuadrática, compararla con un valor constante implica resolver una desigualdad cuadrática.La desigualdad se reorganiza moviendo todos los términos hacia un lado y dividiendo por menos cinco, lo que simplifica el coeficiente principal a uno.Dado que la expresión resultante no se puede factorizar fácilmente, se utiliza la fórmula cuadrática. Los coeficientes se sustituyen en la fórmula para encontrar dos puntos de solución.Estos valores representan el principio y el final del período de tiempo durante el cual el proyectil permanece por encima de los 100 metros.Un gráfico de la función de altura ayuda a visualizar esta situación. La curva se eleva por encima de la marca de los 100 metros, alcanza un pico y luego vuelve a caer a la marca.En el gráfico, la parte de la curva que se encuentra por encima de los 100 metros está sombreada para mostrar el intervalo de tiempo durante el cual se cumple la condición.

desigualdades no lineales

Una desigualdad no lineal describe una comparaci&#243;n que involucra una expresi&#243;n que se curva o se comporta de manera m&#225;s compleja que una recta. Estas desigualdades suelen presentarse en formas que incluyen t&#233;rminos cuadrados, productos o variables en el denominador.Para resolver una desigualdad de este tipo, se comienza por reescribirla de modo que el cero quede en un lado. Por ejemplo, la desigualdad:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:80.44%;'><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>se puede factorizar como:<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>Esta forma facilita la identificaci&#243;n de los valores que hacen que la expresi&#243;n sea igual a cero. En este caso, los valores clave son 3 y -2. Estos valores dividen la recta num&#233;rica en tres intervalos.El solucionador elige un valor de prueba en cada intervalo y lo sustituye en la expresi&#243;n. Esta prueba muestra si la expresi&#243;n en ese tramo es positiva o negativa. Si la desigualdad original exige que la expresi&#243;n sea menor que cero, los intervalos correctos son aquellos en los que el resultado es negativo. La soluci&#243;n completa incluye solo esos intervalos.El mismo enfoque se aplica a las desigualdades racionales, donde la variable aparece en el denominador. Sin embargo, el solucionador tambi&#233;n debe considerar d&#243;nde la expresi&#243;n no est&#225; definida. Por ejemplo:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:95.87%;'><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>En este caso, los valores clave son -1 y 4. Estos dividen la recta num&#233;rica en intervalos separados. El solucionador prueba cada intervalo y excluye el punto en el que el denominador es igual a cero (en este caso, 4).Las desigualdades no lineales son &#250;tiles en la pr&#225;ctica. Un analista empresarial podr&#237;a utilizar una para determinar cu&#225;ndo el beneficio se mantiene mayor que cero. Un ingeniero podr&#237;a aplicarla para garantizar que una m&#225;quina funcione dentro de l&#237;mites de presi&#243;n seguros. Un f&#237;sico podr&#237;a analizar cu&#225;ndo un objeto se mantiene dentro de un cierto rango de movimiento.

Desigualdades no lineales

Una desigualdad no lineal describe una comparaci&#243;n que involucra una expresi&#243;n que se curva o se comporta de manera m&#225;s compleja que una ...

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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

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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...

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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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