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

Em um movimento de projétil vertical sem resistência do ar, um objeto é lançado do nível do solo com uma velocidade de 50 metros por segundo. Sua altura ao longo do tempo segue a equação quadrática: menos cinco t ao quadrado mais cinquenta t, onde menos cinco é metade da aceleração da gravidade. O objetivo é determinar o intervalo de tempo em que a altura do projétil ultrapassa 100 metros.Como a função de altura é quadrática, compará-la com um valor constante envolve a resolução de uma desigualdade quadrática.A desigualdade é reorganizada movendo todos os termos para um lado e dividindo por menos cinco, o que simplifica o coeficiente principal para um.Como a expressão resultante não pode ser fatorada facilmente, a fórmula quadrática é usada. Os coeficientes são substituídos na fórmula para encontrar dois pontos de solução.Esses valores representam o início e o fim do período de tempo durante o qual o projétil permanece acima de 100 metros.Um gráfico da função de altura ajuda a visualizar essa situação. A curva sobe acima da marca de 100 metros, atinge um pico e depois volta à marca.No gráfico, a parte da curva que fica acima de 100 metros é sombreada para mostrar o intervalo de tempo durante o qual a condição é atendida.

desigualdades não lineares

Uma desigualdade n&#227;o linear descreve uma compara&#231;&#227;o envolvendo uma express&#227;o que se curva ou se comporta de maneira mais complexa que uma reta. Essas desigualdades frequentemente aparecem em formas que incluem quadrados, produtos ou vari&#225;veis no denominador.Para resolver tal desigualdade, come&#231;a-se reescrevendo-a de forma que o zero apare&#231;a de um lado. Por exemplo, a desigualdade:<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>pode ser fatorada 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>Essa forma facilita a identifica&#231;&#227;o dos valores que fazem com que a express&#227;o seja igual a zero. Nesse caso, os valores cr&#237;ticos s&#227;o 3 e -2. Esses valores dividem a reta num&#233;rica em tr&#234;s regi&#245;es.O resolvedor escolhe um valor de amostra em cada regi&#227;o e o substitui na express&#227;o. Esse teste mostra se a express&#227;o dentro daquela se&#231;&#227;o &#233; positiva ou negativa. Se a desigualdade original exige que a express&#227;o seja menor que zero, as regi&#245;es corretas s&#227;o aquelas em que o resultado &#233; negativo. A solu&#231;&#227;o completa inclui apenas essas regi&#245;es.A mesma abordagem se aplica a desigualdades racionais, em que a vari&#225;vel aparece no denominador. No entanto, o resolvedor tamb&#233;m deve considerar onde a express&#227;o &#233; indefinida. Por exemplo:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:95.17%;'><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>Neste caso, os valores cr&#237;ticos s&#227;o -1 e 4. Esses valores dividem a reta num&#233;rica em intervalos separados. O resolvedor testa cada intervalo e exclui o ponto onde o denominador se torna zero &#8212; neste caso, 4.Desigualdades n&#227;o lineares s&#227;o &#250;teis em ambientes pr&#225;ticos. Um analista de neg&#243;cios pode us&#225;-las para determinar quando o lucro permanece acima de zero. Um engenheiro pode aplic&#225;-las para garantir que uma m&#225;quina opere dentro de limites de press&#227;o seguros. Um f&#237;sico pode analisar quando um objeto permanece dentro de uma determinada faixa de movimento.

Desigualdades Não Lineares

Uma desigualdade n&#227;o linear descreve uma compara&#231;&#227;o envolvendo uma express&#227;o que se curva ou se comporta de maneira mais complexa que ...

Education

JoVE Core

Math Fundamentals

Fundamentos da Matemática

Unit

Unidade

Polynomial and Rational Functions

Funções Polinomiais e Racionais

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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

solving-nonlinear-inequalities-on-a-number-line

quadratic models

Quadratic Motion and Parabolas

Quadratic models describe relationships where the rate of change changes at a constant rate. In math and science, this often shows up as a curved pattern called a parabola. These models help explain how a system changes over time or under a set force. A common example is the vertical motion of an object moving under gravity, such as a ball thrown into the air. The object rises to a maximum height and then falls back down. Gravity slows the ascent and speeds up the descent, which creates the...

introduction to polynomial functions

Polynomial Graphs, Zeros, and Degree

Polynomial functions are an important part of algebra and calculus. They are made from variables and constants combined by addition, subtraction, and multiplication. The variable is raised to a nonnegative integer exponent. A general polynomial function of degree n has a leading term aₙxⁿ, where aₙ is not equal to 0. The coefficient aₙ is the leading coefficient, and a₀ is the constant term. Polynomials are grouped by degree. A constant function has degree zero, a linear function has degree...

long division of polynomials

Polynomial Division: Quotient and Remainder

Polynomial division breaks a polynomial into a quotient and a remainder. It works like numerical division, but the dividend is a polynomial and the divisor is an algebraic expression. The result shows how much of the divisor fits into the dividend and what is left over. The first step is to write both polynomials in standard form, with terms in descending powers of the variable. Then divide the leading term of the dividend by the leading term of the divisor. That first result becomes the first...

synthetic disvision of polynomials

Using Synthetic Division to Find Factors

Synthetic division is a fast way to divide a polynomial by a linear binomial of the form x - c, where c is a real number. It gives a shorter method than polynomial long division and is useful for checking values and finding factors. The setup starts with the coefficients of the polynomial written in descending order of powers. If a term is missing, a zero must be used to keep the columns lined up. The constant c from the divisor x - c is placed to the left of the synthetic division table. The...

real zeros of polynomials

Finding Polynomial X-Intercepts

Polynomials can be analyzed by finding their real zeros, which are the values of x that make the polynomial equal to zero. These points are also the x-intercepts of the graph. Understanding them helps students see where a polynomial crosses the x-axis. The Rational Zeros Theorem gives a list of possible rational solutions for a polynomial with integer coefficients. If f(x)=anx^n+...+a0, then every rational zero has the form p/q. In that form, p divides the constant term a0 and q divides the...

fundamental theorem of algebra

Polynomial Roots in Complex Numbers

Polynomial roots in complex numbers are a key result in algebra. The Fundamental Theorem of Algebra says that every non-constant polynomial with complex coefficients has at least one complex zero. For a polynomial of degree n, where n is at least 1 and the leading coefficient a n is not 0, there is at least one solution in the complex number system. This result also applies to polynomials with real coefficients, because the real numbers are a subset of the complex numbers. The Complete...

complex zeros

Finding Polynomial Roots With Complex Numbers

Complex numbers help describe every root of a polynomial equation. A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit, with i^2 = -1. A complex zero is any value that makes P(x) = 0 for a polynomial P(x). The complex number system includes the real numbers, so it gives a complete way to study polynomial roots. Every polynomial of degree n  1 can be written as a product of n linear factors over the complex numbers. In that factorization, each c_i...

asymptotes in rational functions

Graphing Rational Functions with Asymptotes

Rational functions are quotients of two polynomials, with the denominator Q(x) not equal to zero. Their graphs often include asymptotes, which are lines the graph gets close to but does not touch. These lines help show how the function behaves near certain input values and at large values of x. A vertical asymptote appears when the denominator is zero and the numerator is not zero. At those x-values, the function is undefined. To find a vertical asymptote, solve Q(x) = 0. For example, a...

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,...

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...

key terms and concepts

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.

math fundamentals

Solving Nonlinear Inequalities on a Number Line

Description

Transcript

Tags