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

Wartość bezwzględna wskazuje odległość, w jakiej liczba znajduje się od zera na osi liczbowej, niezależnie od tego, czy liczba jest dodatnia, czy ujemna.Gdy wartość bezwzględna wielkości jest mniejsza lub równa danej wartości, rozwiązanie obejmuje wszystkie liczby między granicami ujemnymi i dodatnimi.Jeśli wartość bezwzględna jest większa lub równa danej wartości, rozwiązanie obejmuje wszystkie liczby spoza tych granic.Rozwiązania te są pokazane na osi liczbowej za pomocą odcinków lub promieni, z okręgami wskazującymi punkty końcowe uwzględnione ze względu na część równości nierówności.Rozważ zmierzenie prędkości dźwięku za pomocą echa ze znanej odległości od ściany. W zależności od fluktuacji środowiskowych zmierzona prędkość wynosi około 343 metrów na sekundę, z wahaniami plus minus 2 metry na sekundę.Zmienność ta jest modelowana przy użyciu nierówności wartości bezwzględnej, która wyznacza granicę tego, jak daleko obserwowana prędkość może odbiegać od wartości oczekiwanej.Rozwiązanie nierówności polega na odjęciu i dodaniu dozwolonej zmienności do oczekiwanej prędkości, aby uzyskać pełny zakres dopuszczalnych wartości.

nierówności wartości bezwzględnej

Warto&#347;&#263; bezwzgl&#281;dna jest narz&#281;dziem matematycznym, kt&#243;re reprezentuje odleg&#322;o&#347;&#263; liczby od zera na osi liczbowej, niezale&#380;nie od jej znaku. W kontek&#347;cie nier&#243;wno&#347;ci wyra&#380;enia warto&#347;ci bezwzgl&#281;dnej s&#322;u&#380;&#261; do definiowania zakresu dopuszczalnych warto&#347;ci lub granic zmiennej. Nier&#243;wno&#347;ci te s&#261; powszechnie stosowane w modelowaniu naukowym i interpretacji danych, gdzie zmienno&#347;&#263; w obr&#281;bie lub poza okre&#347;lonym progiem musi by&#263; precyzyjnie uj&#281;ta.Nier&#243;wno&#347;&#263; warto&#347;ci bezwzgl&#281;dnej w postaci &#8739;x&#8739; &#8804; a, gdzie a &#8805; 0, opisuje wszystkie warto&#347;ci x od &#8722;a do a w&#322;&#261;cznie. T&#281; nier&#243;wno&#347;&#263; mo&#380;na zapisa&#263; jako nier&#243;wno&#347;&#263; z&#322;o&#380;on&#261;:<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>Graficznie zakres ten przedstawiany jest na osi liczbowej jako odcinek &#322;&#261;cz&#261;cy punkty ko&#324;cowe &#8722;a i a, cz&#281;sto oznaczony domkni&#281;tymi k&#243;&#322;kami, aby pokaza&#263;, &#380;e punkty ko&#324;cowe s&#261; zawarte w rozwi&#261;zaniu.Odwrotnie, nier&#243;wno&#347;&#263; postaci &#8739;x&#8739; &#8805; a oznacza, &#380;e x le&#380;y w odleg&#322;o&#347;ci nie mniejszej ni&#380; a od zera. Ten zbi&#243;r rozwi&#261;za&#324; obejmuje dwa oddzielne przedzia&#322;y:<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_2_2__editor_40036.svg' alt='40036_Equation_2_2_.svg' /></figure>S&#261; one zazwyczaj przedstawiane za pomoc&#261; promieni wychodz&#261;cych z punkt&#243;w ko&#324;cowych, cz&#281;sto oznaczonych domkni&#281;tymi k&#243;&#322;kami, je&#347;li punkty ko&#324;cowe s&#261; uwzgl&#281;dnione w rozwi&#261;zaniu.Rozwa&#380;my reakcj&#281; chemiczn&#261;, kt&#243;ra przebiega optymalnie w temperaturze 80&#176;C. Aby zapewni&#263; bezpiecze&#324;stwo i wydajno&#347;&#263; produktu, temperatura musi mie&#347;ci&#263; si&#281; w odleg&#322;o&#347;ci nie wi&#281;kszej ni&#380; 2,5&#176;C od tej warto&#347;ci. Wymaganie to mo&#380;na zapisa&#263; jako:<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_3_2__editor_40036.svg' alt='40036_Equation_3_2_.svg' /></figure>Ta nier&#243;wno&#347;&#263; oznacza, &#380;e temperatura x nie mo&#380;e odbiega&#263; od warto&#347;ci idealnej o wi&#281;cej ni&#380; 2,5&#176;C. Rozwi&#261;zanie nier&#243;wno&#347;ci daje:<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>Zatem ka&#380;da temperatura w tym przedziale zapewnia optymalne warunki reakcji.

Nierówności wartości bezwzględnej

Warto&#347;&#263; bezwzgl&#281;dna jest narz&#281;dziem matematycznym, kt&#243;re reprezentuje odleg&#322;o&#347;&#263; liczby od zera na osi liczbowej, ...

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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

absolute-value-inequalities-on-a-number-line

real number system

Number Types on the Real Line

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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Simplifying Algebraic Expressions

Algebraic expressions use variables, constants, and operations to show mathematical relationships. They help describe patterns and solve problems across many areas of mathematics. Knowing the parts of an expression makes it easier to simplify and work with it correctly. Each algebraic expression has separate pieces that work together. A coefficient is the number attached to a variable, and an exponent shows repeated multiplication. Some expressions have a single part, while others include...

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Working with Rational Expression Domains

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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Complex Roots of Quadratic Equations

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inequalities

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absolute value inequalities

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...

mathematical modeling: problem solving

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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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