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

Les inégalités comparent deux valeurs pour montrer leur relation à l’aide de symboles tels que inférieur à, supérieur à, inférieur ou égal à, et supérieur ou égal à.Les notations d’intervalle représentent la plage de valeurs qui satisfont une inégalité sous la forme concise d’ensembles de solutions.Un intervalle fermé inclut les extrémités et est indiqué entre crochets, tandis qu’un intervalle ouvert n’inclut pas les extrémités et est indiqué entre parenthèses.Les propriétés des inégalités incluent la règle selon laquelle l’addition d’une valeur ou la soustraction de la même valeur des deux côtés maintient la direction de l’inégalité la même.Multiplier les deux côtés par un nombre positif ou diviser les deux côtés par le même nombre permet également de conserver la même direction, mais multiplier ou diviser par un nombre négatif inverse l’inégalité.Lorsque les deux côtés d’une inégalité sont positifs, l’adoption de réciproques inverse la direction de l’inégalité. Cette règle ne s’applique pas lorsque les valeurs sont négatives.La simplification d’une inégalité consiste à supprimer la constante, puis à diviser les deux côtés par une constante pour isoler la variable.Ces inégalités apparaissent dans la vie réelle. Par exemple, la règle de « plus de 100 cm de hauteur » d’un manège n’autorise que les véhicules de plus de 100 cm.

inégalités

Les in&#233;galit&#233;s expriment des relations math&#233;matiques entre deux valeurs qui ne sont pas &#233;gales et sont compar&#233;es &#224; l&#8217;aide de symboles tels que &lt;, &gt;, &#8804; ou &#8805;. Ces expressions d&#233;finissent un ensemble de solutions possibles plut&#244;t qu&#8217;une valeur unique. La notation par intervalles offre un moyen concis d&#8217;exprimer ces ensembles de solutions, en particulier lorsque la variable prend des valeurs dans un intervalle continu. Un intervalle ouvert, not&#233; (a, b), exclut ses bornes, tandis qu&#8217;un intervalle ferm&#233; [a, b] les inclut. Il existe &#233;galement des intervalles semi-ouverts, tels que (a, b] et [a, b), qui incluent une seule borne. Les notations (a, &#8734;) et (&#8722;&#8734;, b] sont utilis&#233;es lorsqu&#8217;un ensemble s&#8217;&#233;tend ind&#233;finiment dans une direction, les symboles d&#8217;infini n&#8217;&#233;tant jamais consid&#233;r&#233;s comme des valeurs r&#233;elles.
La r&#233;solution des in&#233;galit&#233;s exige l&#8217;application d&#8217;op&#233;rations en respectant des r&#232;gles pr&#233;cises destin&#233;es &#224; maintenir ou &#224; inverser le sens de l&#8217;in&#233;galit&#233;. Lorsqu&#8217;on ajoute ou soustrait une constante aux deux membres, le sens de l&#8217;in&#233;galit&#233; demeure inchang&#233;. Cette r&#232;gle est formalis&#233;e par la propri&#233;t&#233; d&#8217;addition des in&#233;galit&#233;s&#160;: si a &lt; b, alors a + c &lt; b + c. De m&#234;me, multiplier ou diviser les deux membres d&#8217;une in&#233;galit&#233; par un nombre positif conserve son sens, tandis que l&#8217;utilisation d&#8217;un nombre n&#233;gatif l&#8217;inverse. Par exemple, multiplier les deux membres de 3 &lt; 7 par 2 donne 6 &lt; 14, mais multiplier les deux membres de 3 &lt; 7 par &#8722;2 conduit &#224; &#8722;6 &gt; &#8722;14.
La propri&#233;t&#233; de passage &#224; l&#8217;inverse pour les in&#233;galit&#233;s joue &#233;galement un r&#244;le essentiel. Si a &gt; 0, alors 1/a &gt; 0, et prendre les inverses des deux membres d&#8217;une in&#233;galit&#233; portant sur des valeurs strictement positives inverse son sens. Ainsi, si a &lt; b et que les deux valeurs sont strictement positives, alors 1/a &gt; 1/b.

Inégalités

Les in&#233;galit&#233;s expriment des relations math&#233;matiques entre deux valeurs qui ne sont pas &#233;gales et sont compar&#233;es &#224; ...

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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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