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

Eşitsizlikler, daha küçük, daha büyük, daha küçük veya eşit ve daha büyük veya eşit gibi semboller kullanarak ilişkilerini göstermek için iki değeri karşılaştırır.Aralık gösterimleri, bir eşitsizliği karşılayan değer aralığını çözüm kümelerinin kısa bir biçimi olarak temsil eder.Kapalı aralık uç noktaları içerir ve köşeli parantez kullanılarak gösterilirken, açık aralık uç noktaları içermez ve parantez kullanılarak gösterilir.Eşitsizliklerin özellikleri, her iki tarafta da bir değer eklemenin veya aynı değeri çıkarmanın eşitsizliğin yönünü aynı tutması kuralını içerir.Her iki tarafı pozitif bir sayı ile çarpmak veya her iki tarafı da aynı sayıya bölmek de yönü aynı tutar, ancak negatif bir sayı ile çarpmak veya bölmek eşitsizliği tersine çevirir.Bir eşitsizliğin her iki tarafı da pozitif olduğunda, karşılıklı almak eşitsizliğin yönünü tersine çevirir. Değerler negatif olduğunda bu kural geçerli değildir.Bir eşitsizliği basitleştirmek, sabiti kaldırmayı ve ardından değişkeni izole etmek için her iki tarafı bir sabitle bölmeyi içerir.Bu eşitsizlikler gerçek hayatta ortaya çıkar. Örneğin, bir sürüşün '100 cm'den uzun' kuralı yalnızca 100 cm'den uzun olanlara izin verir.

eşitsizlikler

E&#351;itsizlikler, iki de&#287;erin e&#351;it olmad&#305;&#287;&#305;n&#305; ve &lt;, &gt;, &#8804; veya &#8805; gibi sembollerle kar&#351;&#305;la&#351;t&#305;r&#305;ld&#305;&#287;&#305;n&#305; ifade eden matematiksel ili&#351;kilerdir. Bu t&#252;r ifadeler, tek bir say&#305; yerine olas&#305; &#231;&#246;z&#252;mlerden olu&#351;an bir aral&#305;k tan&#305;mlar. Aral&#305;k g&#246;sterimi, &#246;zellikle de&#287;i&#351;kenin s&#252;rekli bir aral&#305;&#287;a yay&#305;ld&#305;&#287;&#305; durumlarda &#231;&#246;z&#252;m k&#252;melerini &#246;zl&#252; bi&#231;imde ifade eder. (a, b) bi&#231;imindeki a&#231;&#305;k aral&#305;k u&#231; noktalar&#305; d&#305;&#351;lar; [a, b] bi&#231;imindeki kapal&#305; aral&#305;k u&#231; noktalar&#305; i&#231;erir. (a, b] ve [a, b) gibi yar&#305; a&#231;&#305;k aral&#305;klar yaln&#305;zca tek bir u&#231; noktay&#305; i&#231;erir. K&#252;menin tek y&#246;nde s&#305;n&#305;rs&#305;z olarak geni&#351;ledi&#287;i durumlarda (a, &#8734;) ve (&#8722;&#8734;, b] gibi g&#246;sterimler kullan&#305;l&#305;r; sonsuzluk sembolleri hi&#231;bir zaman ger&#231;ek de&#287;er olarak k&#252;menin eleman&#305; de&#287;ildir.
E&#351;itsizlikleri &#231;&#246;zerken, e&#351;itsizli&#287;in y&#246;n&#252;n&#252; koruyan ya da tersine &#231;eviren kurallara uygun i&#351;lemler uygulan&#305;r. Her iki tarafa ayn&#305; sabiti eklemek veya &#231;&#305;karmak y&#246;n&#252; de&#287;i&#351;tirmez; bu durum toplama &#246;zelli&#287;iyle ifade edilir: a &lt; b ise a + c &lt; b + c. Benzer bi&#231;imde, her iki taraf&#305; pozitif bir say&#305;yla &#231;arpmak ya da b&#246;lmek y&#246;n&#252; korur; negatif bir say&#305;yla &#231;arpma ya da b&#246;lme ise y&#246;n&#252; tersine &#231;evirir. &#214;rne&#287;in, 3 &lt; 7 e&#351;itsizli&#287;inin her iki taraf&#305;n&#305; 2 ile &#231;arpmak 6 &lt; 14 sonucunu verirken, ayn&#305; e&#351;itsizli&#287;in her iki taraf&#305;n&#305; &#8722;2 ile &#231;arpmak &#8722;6 &gt; &#8722;14 sonucunu verir.
E&#351;itsizliklerde &#231;arpma tersi (resiprokal) &#246;zelli&#287;i de &#246;nemlidir. a &gt; 0 ise 1/a &gt; 0&#8217;d&#305;r ve yaln&#305;zca pozitif de&#287;erler s&#246;z konusu oldu&#287;unda her iki taraf&#305;n &#231;arpma tersini almak e&#351;itsizli&#287;in y&#246;n&#252;n&#252; tersine &#231;evirir. Nitekim a &lt; b ve a, b &gt; 0 ise 1/a &gt; 1/b elde edilir.

Eşitsizlikler

E&#351;itsizlikler, iki de&#287;erin e&#351;it olmad&#305;&#287;&#305;n&#305; ve &lt;, &gt;, &#8804; veya &#8805; gibi sembollerle ...

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Math Fundamentals

Matematik Temelleri

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Foundations of Mathematics

Matematiğin Temelleri

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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

real number system

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The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two...

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Radicals show the root of a number and the exponent needed to get that value. A radical expression has two main parts. The radicand is the number inside the root symbol, and the index tells which root to take. The notation n√a means the principal nth root of a. The index changes the type of root. When n equals 2, the expression is a square root. When n equals 3, it is a cube root. The sign of the radicand also matters. If the index is even, a negative radicand does not give a real result. If...

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Rational expressions are algebraic fractions with polynomials in the numerator and denominator. They follow the same basic rules as numerical fractions, but variables make them more delicate to work with. A key step is finding values that make the expression undefined, usually because of division by zero or an undefined radical. To determine the domain, start by identifying the values excluded from the denominator. The domain includes all real numbers except those that make the denominator...

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quadratic-equations-in-the-complex-number-system

Complex Roots of Quadratic Equations

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inequalities

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...

absolute value inequalities

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mathematical modeling: problem solving

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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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