Name: solving nonlinear inequalities on a number line
Uploaded: 2025-11-26T05:43:48.000Z
Duration: 1 min 29 s
Channel: JoVE

W pionowym ruchu pocisku bez oporu powietrza obiekt jest wystrzeliwany z poziomu gruntu z prędkością 50 metrów na sekundę. Jego wysokość w czasie jest zgodna z równaniem kwadratowym: minus pięć t do kwadratu plus pięćdziesiąt t, gdzie minus pięć to połowa przyspieszenia ziemskiego. Celem jest określenie przedziału czasowego, w którym wysokość pocisku przekracza 100 metrów.Ponieważ funkcja wysokości jest kwadratowa, porównanie jej ze stałą wartością wymaga rozwiązania nierówności kwadratowej.Nierówność jest przestawiana poprzez przesunięcie wszystkich wyrazów na jedną stronę i podzielenie przez minus pięć, co upraszcza współczynnik wiodący do jednego.Ponieważ wynikowego wyrażenia nie można łatwo rozłożyć na czynniki pierwsze, używana jest formuła kwadratowa. Współczynniki są podstawiane do wzoru, aby znaleźć dwa punkty rozwiązania.Wartości te reprezentują początek i koniec okresu, w którym pocisk utrzymuje się na wysokości powyżej 100 metrów.Wykres funkcji wysokości pomaga zobrazować tę sytuację. Krzywa wznosi się powyżej znaku 100 metrów, osiąga szczyt, a następnie opada z powrotem do znaku.Na wykresie część krzywej, która leży powyżej 100 metrów, jest zacieniona, aby pokazać przedział czasu, w którym warunek jest spełniony.

nierówności nieliniowe

Nier&#243;wno&#347;&#263; nieliniowa opisuje por&#243;wnanie, w kt&#243;rym wyra&#380;enie ma przebieg krzywoliniowy lub zachowuje si&#281; w spos&#243;b bardziej z&#322;o&#380;ony ni&#380; prosta. Nier&#243;wno&#347;ci te cz&#281;sto pojawiaj&#261; si&#281; w postaciach zawieraj&#261;cych kwadraty, iloczyny lub zmienne w mianowniku.Aby rozwi&#261;za&#263; tak&#261; nier&#243;wno&#347;&#263;, nale&#380;y najpierw przekszta&#322;ci&#263; j&#261; tak, aby po jednej stronie znajdowa&#322;o si&#281; zero. Na przyk&#322;ad nier&#243;wno&#347;&#263;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:80.09%;'><img style='aspect-ratio:1920/74;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_1_editor_40060.svg' alt='40060_Equation_1.svg' /></figure>mo&#380;na roz&#322;o&#380;y&#263; na czynniki do postaci:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:96.22%;'><img style='aspect-ratio:300/9;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_2_editor_40060.svg' alt='40060_Equation_2.svg' /></figure>Taka posta&#263; u&#322;atwia wskazanie warto&#347;ci, dla kt&#243;rych wyra&#380;enie przyjmuje warto&#347;&#263; zero. W tym przypadku punktami kluczowymi s&#261; 3 oraz -2. Warto&#347;ci te dziel&#261; o&#347; liczbow&#261; na trzy przedzia&#322;y.W ka&#380;dym przedziale wybiera si&#281; przyk&#322;adow&#261; warto&#347;&#263; i podstawia do wyra&#380;enia. Taki test pokazuje, czy wyra&#380;enie w danym przedziale jest dodatnie, czy ujemne. Je&#347;li pierwotna nier&#243;wno&#347;&#263; wymaga, by wyra&#380;enie by&#322;o mniejsze od zera, poprawnymi przedzia&#322;ami s&#261; te, w kt&#243;rych wynik jest ujemny. Ostateczne rozwi&#261;zanie obejmuje wy&#322;&#261;cznie te przedzia&#322;y.To samo podej&#347;cie stosuje si&#281; do nier&#243;wno&#347;ci wymiernych, w kt&#243;rych zmienna wyst&#281;puje w mianowniku. Nale&#380;y jednak dodatkowo uwzgl&#281;dni&#263; miejsca, w kt&#243;rych wyra&#380;enie jest nieokre&#347;lone. Na przyk&#322;ad:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:96.92%;'><img style='aspect-ratio:300/20;' src='https://cloudfront.jove.com/files/ftp_upload/40060/40060_Equation_3_editor_40060.svg' alt='40060_Equation_3.svg' /></figure>W tym przypadku warto&#347;ciami kluczowymi s&#261; -1 oraz 4. Dziel&#261; one o&#347; liczbow&#261; na osobne przedzia&#322;y. Nale&#380;y przetestowa&#263; ka&#380;dy przedzia&#322; i wykluczy&#263; punkt, w kt&#243;rym mianownik staje si&#281; zerem &#8211; w tym przypadku 4.Nier&#243;wno&#347;ci nieliniowe znajduj&#261; zastosowanie w praktyce. Analityk biznesowy mo&#380;e wykorzysta&#263; je do okre&#347;lenia, kiedy zysk pozostaje powy&#380;ej zera. In&#380;ynier mo&#380;e zastosowa&#263; je, aby upewni&#263; si&#281;, &#380;e maszyna pracuje w bezpiecznych zakresach ci&#347;nienia. Fizyk mo&#380;e bada&#263;, kiedy obiekt pozostaje w okre&#347;lonym zakresie ruchu.

Nierówności nieliniowe

Nier&#243;wno&#347;&#263; nieliniowa opisuje por&#243;wnanie, w kt&#243;rym wyra&#380;enie ma przebieg krzywoliniowy lub zachowuje si&#281; w spos&#243;b ...

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Polynomial and Rational Functions

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KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

Application of Nonlinear Inequalities

solving nonlinear inequalities on a number line

Nonlinear inequalities can be solved by checking where an expression is positive or negative on a number line. These inequalities often include squares, products, or variables in the denominator, so they do not follow the same simple pattern as linear inequalities. The first step is to rewrite the inequality so that zero is on one side.

After rewriting, factor the expression if possible. Factoring helps you find the key values where the expression equals zero. In one example, the values 3 and -2 split the number line into three regions. A sample value from each region is then tested in the expression to see whether that interval gives a positive or negative result.

The final answer includes only the regions that match the original sign of the inequality. If the inequality says the expression must be less than zero, then only the intervals with negative results are included. The same interval test works for rational inequalities, where the variable appears in the denominator. These problems also require checking where the expression is undefined, so any value that makes the denominator zero must be left out.

For example, a rational inequality may have key values of -1 and 4. These values divide the number line into separate intervals. Each interval is tested, but the point where the denominator becomes zero, such as 4, is excluded from the solution. This process is useful in real situations, such as finding when profit stays above zero, keeping machine pressure within safe limits, or checking when an object stays within a certain range of motion.

Solving Nonlinear Inequalities on a Number Line

solving-nonlinear-inequalities-on-a-number-line

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application of nonlinear inequalities

Nonlinear inequalities can be solved by checking where an expression is positive or negative on a number line. These inequalities often include squares, products, or variables in the denominator, so they do not follow the same simple pattern as linear inequalities. The first step is to rewrite the inequality so that zero is on one side. After rewriting, factor the expression if possible. Factoring helps you find the key values where the expression equals zero. In one example, the values 3 and...

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Learn how to solve nonlinear and rational inequalities by factoring, testing number line intervals, and excluding undefined values in grade 9–12 math.

Solving Nonlinear 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.

math fundamentals

Solving Nonlinear Inequalities on a Number Line

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