Name: graphing and solving nonlinear inequalities
Uploaded: 2025-11-26T05:43:49.000Z
Duration: 1 min 25 s
Channel: JoVE

En la desigualdad lineal, el límite describe una línea recta que separa la región de solución de la región que satisface la otra condición de la desigualdad.Por el contrario, una desigualdad no lineal implica expresiones de mayor grado y define regiones utilizando límites curvos en lugar de líneas rectas.Cuando se trabaja con tales desigualdades, particularmente los tipos polinómicos, las soluciones a menudo se encuentran factorizando la expresión y estableciendo cada factor igual a cero para determinar las raíces o ceros.Estos puntos no se incluyen en la solución, pero indican dónde cambia de signo la expresión, dividiendo la recta numérica en intervalos que se pueden probar para determinar su validez dentro de la desigualdad.Un valor de cada intervalo se sustituye por la desigualdad original para determinar si la expresión es verdadera en esa región.Los intervalos en los que se satisface la desigualdad definen las regiones de solución.En un contexto empresarial, el beneficio es la diferencia entre los ingresos y el coste de las unidades vendidas. Para asegurar una ganancia de al menos 20 dólares, la condición se expresa como una desigualdad no lineal. Resolver esta desigualdad da el rango específico de unidades para las que se satisface el requisito de ganancias.

introducción a las desigualdades no lineales

Las desigualdades lineales y no lineales son fundamentales para analizar las relaciones entre variables e identificar rangos que cumplen condiciones espec&#237;ficas. Una desigualdad lineal implica variables elevadas solo a la primera potencia, lo que da lugar a una recta. Esta recta divide el plano de coordenadas en dos regiones distintas: una que satisface la desigualdad y otra que no. Cada regi&#243;n representa un conjunto de soluciones en el que la relaci&#243;n lineal se cumple bajo la restricci&#243;n especificada.
Las desigualdades no lineales se diferencian por involucrar variables elevadas a potencias mayores que uno. Estas incluyen desigualdades cuadr&#225;ticas y polin&#243;micas de grado superior, que producen gr&#225;ficas curvas. Dichas curvas representan relaciones en las que la tasa de cambio no es constante, sino que var&#237;a con el valor de entrada. Resolver estas desigualdades implica examinar c&#243;mo se comporta la funci&#243;n en diferentes intervalos de la recta num&#233;rica.
Los puntos en los que la expresi&#243;n es igual a cero, o ra&#237;ces, indican los puntos donde la funci&#243;n puede cambiar de signo. Estos puntos dividen la recta num&#233;rica en intervalos, cada uno de los cuales puede verificarse sustituyendo un valor representativo en la desigualdad original. Los intervalos donde se cumple la desigualdad conforman el conjunto de soluciones.
En entornos aplicados como la econom&#237;a o los negocios, las desigualdades no lineales aportan informaci&#243;n valiosa. Por ejemplo, una funci&#243;n de beneficio influida por el volumen de producci&#243;n y los costes marginales puede modelarse como una expresi&#243;n no lineal. Al aplicar la desigualdad, es posible identificar el rango de ventas que garantiza que el beneficio se mantenga por encima de un umbral espec&#237;fico, como 500 d&#243;lares estadounidenses, lo que orienta eficazmente la toma de decisiones basada en el an&#225;lisis matem&#225;tico.

Introducción a las desigualdades no lineales

Las desigualdades lineales y no lineales son fundamentales para analizar las relaciones entre variables e identificar rangos que cumplen condiciones ...

Education

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JoVE Core

Math Fundamentals

Fundamentos de matemáticas

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Unidad

Polynomial and Rational Functions

Funciones polinómicas y racionales

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Introduction to Nonlinear Inequalities

graphing and solving nonlinear inequalities

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, not straight. The changing shape shows that the rate of change is not constant as the input changes.

Solving a nonlinear inequality often means looking at intervals on the number line. Critical points are the values where the expression equals zero. These points show where the function may change sign. They split the number line into sections that can be tested with a representative value from each interval.

The intervals that make the inequality true form the solution set. This approach is useful in applied settings such as economics and business. A profit function can depend on production volume and marginal costs. By using a nonlinear inequality, it is possible to find the sales range that keeps profit above a target such as five hundred dollars.

Graphing and Solving Nonlinear Inequalities

graphing-and-solving-nonlinear-inequalities

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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,...

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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...

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Learn how linear and nonlinear inequalities differ, how critical points split a number line, and how interval testing finds the solution set.

Graphing and Solving Nonlinear Inequalities | 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

Graphing and Solving Nonlinear Inequalities

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