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

Las desigualdades comparan dos valores para mostrar su relación usando símbolos como menor que, mayor que, menor o igual que y mayor o igual que.Las notaciones de intervalo representan el intervalo de valores que satisfacen una desigualdad como una forma concisa de conjuntos de soluciones.Un intervalo cerrado incluye los puntos finales y se muestra entre corchetes, mientras que un intervalo abierto no incluye los puntos finales y se muestra entre paréntesis.Las propiedades de las desigualdades incluyen la regla de que agregar un valor o restar el mismo valor en ambos lados mantiene la misma dirección de la desigualdad.Multiplicar ambos lados por un número positivo o dividir ambos lados por el mismo número también mantiene la misma dirección, pero multiplicar o dividir por un número negativo invierte la desigualdad.Cuando ambos lados de una desigualdad son positivos, tomar recíprocos invierte la dirección de la desigualdad. Esta regla no se aplica cuando los valores son negativos.Simplificar una desigualdad implica eliminar la constante y luego dividir ambos lados con una constante para aislar la variable.Estas desigualdades aparecen en la vida real. Por ejemplo, la regla de "más de 100 cm de altura" de una atracción solo permite a aquellos que miden más de 100 cm.

desigualdades

Las desigualdades expresan relaciones matem&#225;ticas en las que dos valores no son iguales y se comparan mediante s&#237;mbolos como &lt;, &gt;, &#8804; y &#8805;. Estas expresiones definen un rango de posibles soluciones en lugar de un &#250;nico valor. La notaci&#243;n de intervalos proporciona una forma concisa de expresar estos conjuntos de soluciones, especialmente cuando la variable abarca un rango continuo. Un intervalo abierto, escrito como (a, b), excluye los extremos o l&#237;mites, mientras que un intervalo cerrado [a, b] los incluye. Tambi&#233;n existen intervalos semiabiertos o semicerrados, como (a, b] y [a, b), que incluyen solo un extremo. Las notaciones (a, &#8734;) y (&#8722;&#8734;, b] se utilizan cuando un conjunto se extiende indefinidamente en una direcci&#243;n, y los s&#237;mbolos de infinito nunca se incluyen como valores reales.
Resolver desigualdades requiere aplicar operaciones respetando reglas espec&#237;ficas que mantienen o invierten la direcci&#243;n de la desigualdad. Cuando se suma o resta una constante a ambos lados, la direcci&#243;n de la desigualdad permanece inalterada. Esto se formaliza mediante la propiedad aditiva: si a &lt; b, entonces a + c &lt; b + c. De manera similar, multiplicar o dividir ambos lados por un n&#250;mero positivo conserva la direcci&#243;n de la desigualdad, mientras que usar un n&#250;mero negativo la invierte. Por ejemplo, multiplicar ambos lados de 3 &lt; 7 por 2 da como resultado 6 &lt; 14, pero multiplicar ambos lados de 3 &lt; 7 por &#8722;2 da como resultado &#8722;6 &gt; &#8722;14.
La propiedad rec&#237;proca de las desigualdades tambi&#233;n desempe&#241;a un papel crucial. Si a &gt; 0, entonces 1/a &gt; 0, y al tomar los valores rec&#237;procos de ambos lados de una desigualdad con valores positivos se invierte la direcci&#243;n de la desigualdad. Por ejemplo, si a &lt; b y ambos son positivos, entonces 1/a &gt; 1/b.

Desigualdades

Las desigualdades expresan relaciones matem&#225;ticas en las que dos valores no son iguales y se comparan mediante s&#237;mbolos como &lt;, &gt;, &#8804; ...

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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

solving-inequalities-and-interval-notation

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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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