Name: solving inequalities and interval notation
Uploaded: 2025-11-26T05:44:26.000Z
Duration: 1 min 28 s
Channel: JoVE

As inequações comparam dois valores para mostrar sua relação usando símbolos como menor que, maior que, menor ou igual a e maior ou igual a.As notações de intervalo representam o intervalo de valores que satisfazem uma desigualdade como uma forma concisa de conjuntos de soluções.Um intervalo fechado inclui os pontos finais e é mostrado entre colchetes, enquanto um intervalo aberto não inclui os pontos finais e é mostrado usando parênteses.As propriedades das desigualdades incluem a regra de que adicionar um valor ou subtrair o mesmo valor em ambos os lados mantém a mesma direção da desigualdade.Multiplicar ambos os lados por um número positivo ou dividir ambos os lados pelo mesmo número também mantém a mesma direção, mas multiplicar ou dividir por um número negativo inverte a desigualdade.Quando ambos os lados de uma desigualdade são positivos, tomar recíprocos inverte a direção da desigualdade. Essa regra não se aplica quando os valores são negativos.Simplificar uma desigualdade envolve remover a constante e, em seguida, dividir ambos os lados com uma constante para isolar a variável.Essas desigualdades aparecem na vida real. Por exemplo, a regra de 'mais de 100 cm de altura' de um passeio permite apenas aqueles com mais de 100 cm de altura.

desigualdades

Desigualdades expressam rela&#231;&#245;es matem&#225;ticas em que dois valores n&#227;o s&#227;o iguais e s&#227;o comparados usando s&#237;mbolos como &lt;, &gt;, &#8804; ou &#8805;. Essas express&#245;es definem um conjunto de solu&#231;&#245;es poss&#237;veis, em vez de um &#250;nico valor. A nota&#231;&#227;o de intervalo fornece uma forma concisa de representar esses conjuntos de solu&#231;&#245;es, especialmente quando a vari&#225;vel percorre um intervalo cont&#237;nuo. Um intervalo aberto, escrito como (a, b), exclui os extremos (limites), enquanto um intervalo fechado [a, b] os inclui. Existem tamb&#233;m intervalos semiabertos, como (a, b] e [a, b), que incluem apenas um dos extremos. As nota&#231;&#245;es (a, &#8734;) e (&#8722;&#8734;, b] s&#227;o usadas quando um conjunto se estende indefinidamente em uma dire&#231;&#227;o, sendo que os s&#237;mbolos de infinito nunca s&#227;o considerados valores reais.
Resolver desigualdades requer aplicar opera&#231;&#245;es de acordo com regras espec&#237;ficas que preservam ou invertem a dire&#231;&#227;o da desigualdade. Quando uma constante &#233; adicionada ou subtra&#237;da em ambos os lados, a dire&#231;&#227;o da desigualdade permanece inalterada. Isso &#233; formalizado na propriedade da adi&#231;&#227;o: se a &lt; b, ent&#227;o a + c &lt; b + c. Da mesma forma, multiplicar ou dividir ambos os lados por um n&#250;mero positivo mant&#233;m a dire&#231;&#227;o da desigualdade, enquanto ao usar um n&#250;mero negativo, ela se inverte. Por exemplo, multiplicar ambos os lados de 3 &lt; 7 por 2 resulta em 6 &lt; 14, mas multiplicar ambos os lados de 3 &lt; 7 por &#8722;2 resulta em &#8722;6 &gt; &#8722;14.
A propriedade rec&#237;proca das desigualdades tamb&#233;m desempenha um papel fundamental. Se a &gt; 0, ent&#227;o 1/a &gt; 0, e ao tomar os rec&#237;procos de ambos os lados de uma desigualdade envolvendo valores positivos, inverte-se a dire&#231;&#227;o da desigualdade. Por exemplo, se a &lt; b e ambos s&#227;o positivos, ent&#227;o 1/a &gt; 1/b.

Desigualdades

Desigualdades expressam rela&#231;&#245;es matem&#225;ticas em que dois valores n&#227;o s&#227;o iguais e s&#227;o comparados usando s&#237;mbolos como ...

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Inequalities

solving inequalities and interval notation

Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value.

Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint.

Intervals can also extend forever in one direction. The forms (a, ∞) and (−∞, b] are used for these cases. The infinity symbols do not count as actual values.

Solving inequalities depends on keeping the inequality sign or reversing it at the right time. If the same constant is added or subtracted from both sides, the direction stays the same. This is the addition property: if a −14 when both sides are multiplied by −2.

The reciprocal property is also important for positive values. If a &gt; 0, then 1/a &gt; 0. For positive numbers, taking reciprocals reverses the inequality direction. So if a 1/b.

Solving Inequalities and Interval Notation

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Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value. Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint. Intervals can also extend forever in one...

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key terms and concepts

Learn how inequalities, interval notation, and inequality rules work. See how open, closed, and half-open intervals describe solution sets.

Solving Inequalities and Interval Notation | 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 Inequalities and Interval Notation

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