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

O valor absoluto indica a distância que um número está de zero na reta numérica, independentemente de o número ser positivo ou negativo.Quando o valor absoluto de uma quantidade é menor ou igual a um determinado valor, a solução inclui todos os números entre os limites negativo e positivo.Se o valor absoluto for maior ou igual a um determinado valor, a solução incluirá todos os números fora desses limites.Essas soluções são mostradas em uma reta numérica usando segmentos ou raios, com círculos indicando pontos finais incluídos devido à parte de igualdade da desigualdade.Considere medir a velocidade do som usando ecos de uma distância conhecida de uma parede. Dependendo das flutuações ambientais, a velocidade medida é de cerca de 343 metros por segundo, com uma variação de mais ou menos 2 metros por segundo.Essa variação é modelada usando uma desigualdade de valor absoluto, que define um limite de até que ponto a velocidade observada pode se desviar do valor esperado.Resolver a desigualdade envolve subtrair e adicionar a variação permitida à velocidade esperada para obter a faixa completa de valores aceitáveis.

desigualdades de valor absoluto

O valor absoluto &#233; uma ferramenta matem&#225;tica que representa a dist&#226;ncia de um n&#250;mero em rela&#231;&#227;o a zero na reta real, independentemente do seu sinal. No contexto das desigualdades, as express&#245;es de valor absoluto ajudam a definir intervalos ou limites de valores permitidos para uma vari&#225;vel. Essas desigualdades s&#227;o amplamente utilizadas em modelagem cient&#237;fica e na interpreta&#231;&#227;o de dados, quando &#233; necess&#225;rio representar com precis&#227;o a variabilidade dentro ou al&#233;m de um determinado limite.Uma desigualdade de valor absoluto da forma &#8739;x&#8739; &#8804; a, onde a &#8805; 0, descreve todos os valores de x entre &#8722;a e a, inclusive. Essa desigualdade pode ser reescrita como uma desigualdade composta:<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_1_2__editor_40036.svg' alt='40036_Equation_1_2_.svg' /></figure>Graficamente, esse intervalo &#233; representado em uma reta real por um segmento cont&#237;nuo conectando os extremos em &#8722;a e a, geralmente marcado com c&#237;rculos fechados para indicar que os pontos extremos est&#227;o inclu&#237;dos na solu&#231;&#227;o.Por outro lado, uma desigualdade da forma &#8739;x&#8739; &#8805; a indica que x est&#225; a uma dist&#226;ncia m&#237;nima de pelo menos a unidades em rela&#231;&#227;o a zero. Esse conjunto de solu&#231;&#245;es compreende dois intervalos distintos:<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_2_2__editor_40036.svg' alt='40036_Equation_2_2_.svg' /></figure>Esses intervalos s&#227;o normalmente representados por raios que se estendem para fora a partir dos extremos, frequentemente marcados com c&#237;rculos fechados quando os extremos est&#227;o inclu&#237;dos.Considere uma rea&#231;&#227;o qu&#237;mica que ocorre de maneira ideal a 80&#176;C. Para garantir a seguran&#231;a e o rendimento do produto final, a temperatura deve permanecer dentro de 2,5&#176;C desse valor. Essa condi&#231;&#227;o pode ser expressa como:<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_3_2__editor_40036.svg' alt='40036_Equation_3_2_.svg' /></figure>Essa desigualdade indica que a temperatura x pode variar at&#233; 2,5&#176;C em torno do valor ideal. Resolvendo a desigualdade, obt&#233;m-se:<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_4_2__editor_40036.svg' alt='40036_Equation_4_2_.svg' /></figure>Portanto, qualquer temperatura dentro desse intervalo mant&#233;m as condi&#231;&#245;es &#243;timas da rea&#231;&#227;o.

Desigualdades de Valor Absoluto

O valor absoluto &#233; uma ferramenta matem&#225;tica que representa a dist&#226;ncia de um n&#250;mero em rela&#231;&#227;o a zero na reta real, ...

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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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