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

In un movimento verticale del proiettile senza resistenza dell'aria, un oggetto viene lanciato dal livello del suolo con una velocità di 50 metri al secondo. La sua altezza nel tempo segue l'equazione quadratica: cinque t al quadrato più cinquanta t, dove cinque negativo è la metà dell'accelerazione dovuta alla gravità. L'obiettivo è determinare l'intervallo di tempo in cui l'altezza del proiettile supera i 100 metri.Poiché la funzione di altezza è quadratica, il confronto con un valore costante implica la risoluzione di una disuguaglianza quadratica.La disuguaglianza viene riorganizzata spostando tutti i termini da un lato e dividendo per cinque negativi, il che semplifica il coefficiente iniziale a uno.Poiché l'espressione risultante non può essere fattorizzata facilmente, viene utilizzata la formula quadratica. I coefficienti vengono sostituiti nella formula per trovare due punti di soluzione.Questi valori rappresentano l'inizio e la fine del periodo di tempo durante il quale il proiettile rimane sopra i 100 metri.Un grafico della funzione di altezza aiuta a visualizzare questa situazione. La curva supera i 100 metri, raggiunge un picco e poi ricade verso il segno.Sul grafico, la parte della curva che si trova al di sopra dei 100 metri è ombreggiata per mostrare l'intervallo di tempo durante il quale la condizione è soddisfatta.

disequazioni non lineari

Una disequazione non lineare descrive un confronto che coinvolge un'espressione che curva o si comporta in modo pi&#249; complesso di una retta. Queste disequazioni spesso si presentano in forme che includono termini al quadrato, prodotti o variabili al denominatore.Per risolvere una disequazione di questo tipo, si riscrive in modo che lo zero compaia su un lato. Ad esempio, la disequazione:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><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>pu&#242; essere scomposta in fattori come:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:100%;'><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>Questa forma facilita l'identificazione dei valori per cui l'espressione si annulla. In questo caso, i valori chiave sono 3 e -2. Questi valori dividono la retta numerica in tre regioni.Si sceglie un valore di prova in ciascuna regione e lo si sostituisce nell'espressione. Questo test indica se l'espressione in quell'intervallo &#232; positiva o negativa. Se la disequazione originale richiede che l'espressione sia minore di zero, le regioni corrette sono quelle in cui il risultato &#232; negativo. La soluzione completa include solo quelle regioni.Lo stesso approccio si applica alle disequazioni razionali, in cui la variabile compare al denominatore. Tuttavia, si devono anche considerare i punti in cui l'espressione &#232; indefinita. Ad esempio:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:100%;'><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>In questo caso, i valori chiave sono -1 e 4. Questi dividono la retta numerica in intervalli separati. Si verifica ogni intervallo e si esclude il punto in cui il denominatore si annulla, ossia 4.Le disequazioni non lineari sono utili in contesti pratici. Un analista aziendale potrebbe usarne una per determinare quando il profitto &#232; superiore a zero. Un ingegnere potrebbe applicarne una per garantire che una macchina funzioni entro i limiti di sicurezza della pressione. Un fisico potrebbe analizzare quando un oggetto rimane entro un certo intervallo di moto.

Disequazioni non lineari

Una disequazione non lineare descrive un confronto che coinvolge un'espressione che curva o si comporta in modo pi&#249; complesso di una retta. Queste ...

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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

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Linear and nonlinear inequalities help describe variable relationships and the ranges that satisfy a condition. A linear inequality uses variables raised only to the first power. Its graph is a straight line that divides the coordinate plane into two regions. One region satisfies the inequality, and the other does not. Nonlinear inequalities use variables raised to powers greater than one. These include quadratic inequalities and higher-degree polynomial inequalities. Their graphs are curved,...

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.

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Solving Nonlinear Inequalities on a Number Line

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