Materie

Matematica

Relazioni e funzioni

Funzioni goniometriche

Formule addizione sottrazione, duplicazione, bisezione

Scopri le Equazioni e Disequazioni con Valore Assoluto: Esercizi e Schemi Facili

Name: Knowunity
Brand: Knowunity

Vedi

Scopri le Equazioni e Disequazioni con Valore Assoluto: Esercizi e Schemi Facili

Lele☁️🥥

@walkerele

371 Follower

Segui

Le equazioni con valori assoluti e la loro risoluzione sono argomenti fondamentali in algebra. Questo documento fornisce una guida completa su come affrontare tali equazioni, inclusa la loro interpretazione grafica e le tecniche per risolvere disequazioni con valori assoluti. Vengono trattati anche casi speciali come equazioni irrazionali.

Punti chiave:

Definizione e proprietà dei valori assoluti
Metodi di risoluzione per equazioni con uno o più valori assoluti
Interpretazione grafica delle equazioni con valori assoluti
Tecniche per risolvere disequazioni con valori assoluti
Approccio alle equazioni irrazionali

18/1/2023

9331

Equazioni con valori accolati
(x) =
●
• Ixl l-xl VxER
|x+41 |x|+ [4]
•|-x-y|=1-x)-141
●
●
X, Sex30
-X, se X 20
|x:41=1x1:141
• 1x1 = lul
|x|

Vedi

Page 4: Synthetic Equations and General Strategies

This page covers synthetic equations involving absolute values and presents general strategies for solving absolute value equations.

Key points:

Handling equations where the absolute value term is set equal to a quadratic or other non-linear expression
General principles for solving |A(x)| = K and |A(x)| = |B(x)| type equations
A comprehensive summary schema for approaching absolute value equations

Example: The equation |x² - 9| + |x - 3| = 0 is solved, illustrating how to handle the sum of two absolute value terms.

Highlight: The general strategy involves studying the sign of expressions inside absolute value symbols and considering different cases based on possible combinations.

The page concludes with a useful summary schema that outlines the steps for solving various types of absolute value equations, providing a valuable reference for students.

Vedi

Page 1: Equations with Absolute Values

This page introduces key concepts and formulas for working with absolute value equations.

Key points:

The absolute value of a number is its distance from zero on a number line
Absolute value equations often require considering multiple cases
Common absolute value identities and properties are presented

Definition: The absolute value of x, denoted |x|, is defined as x if x ≥ 0 and -x if x < 0.

Example: |x + 4| = |x| + 4 demonstrates how absolute values interact with addition.

Highlight: When solving |A(x)| = B(x), consider the cases A(x) = B(x) and A(x) = -B(x) separately.

The page provides several solved examples of absolute value equations, illustrating different solution techniques and potential pitfalls to avoid.

Vedi

Page 6: Advanced Absolute Value Inequalities

This page delves deeper into absolute value inequalities, covering more complex scenarios and general solution strategies.

Key topics:

General forms of absolute value inequalities and their solutions
Handling inequalities with multiple absolute value terms
Strategies for solving |A(x)| < |B(x)| type inequalities

Definition: For K > 0, |x| < K is equivalent to -K < x < K, while |x| > K is equivalent to x < -K or x > K.

Example: The inequality |x² - 4| < 5 is solved, demonstrating how to handle quadratic expressions within absolute value symbols.

The page emphasizes the importance of understanding the behavior of absolute value functions and how this knowledge can be applied to solve complex inequalities efficiently.

Vedi

Page 5: Inequalities with Absolute Values

This page introduces inequalities involving absolute values, a crucial topic in advanced algebra.

Key concepts covered:

Solving inequalities of the form |x - a| < b and |x - a| > b
Handling compound inequalities with absolute value terms
Graphical interpretation of absolute value inequalities

Example: The inequality |5x - 1| < 3x + 2 is solved step-by-step, demonstrating how to handle cases and combine solution intervals.

Highlight: When solving absolute value inequalities, it's essential to consider the direction of the inequality and how it affects the solution process.

The page provides multiple solved examples, each illustrating different techniques and potential complexities that can arise in absolute value inequalities.

Vedi

Page 7: Irrational Equations

This page introduces irrational equations, focusing on equations involving square roots.

Key concepts:

Solving equations of the form √A(x) = B(x)
Importance of checking domain restrictions and extraneous solutions
Handling equations with higher index roots

Highlight: When solving irrational equations, it's crucial to check for extraneous solutions by substituting results back into the original equation.

Example: The equation √3x + 4 = 2 + x is solved step-by-step, demonstrating the process of squaring both sides and checking solutions.

The page also covers more complex irrational equations, including those with cube roots and nested radicals, providing a comprehensive overview of solution techniques.

Vedi

Page 2: Graphical Interpretation of Absolute Value Equations

This page explores the graphical representation of absolute value equations and inequalities.

Key concepts covered:

Visualizing absolute value functions on a coordinate plane
Using graphs to understand solution behavior
Interpreting intersection points and regions

Example: The equation |x - 5| = 3x - 1 is analyzed graphically, showing how the solution x = 3/2 corresponds to the intersection point of two functions.

The page also covers more complex equations like |3x - 5| = 2x + 1, demonstrating how to handle piecewise functions and multiple solution intervals.

Highlight: Graphical analysis can provide valuable insights into the number and nature of solutions for absolute value equations and inequalities.

Vedi

Page 8: Advanced Irrational Equations

This page covers more advanced topics in irrational equations, including systems with multiple radicals and irrational inequalities.

Key topics:

Solving equations with multiple radical terms
Techniques for handling irrational inequalities
Graphical interpretation of irrational equations and inequalities

Example: The equation 2√(2+x) - √(x-3) = 4 is solved, demonstrating how to handle equations with multiple radical terms.

Highlight: When solving irrational inequalities, it's important to consider domain restrictions and the behavior of radical functions.

The page concludes with strategies for solving complex irrational equations and inequalities, emphasizing the importance of systematic approaches and careful analysis of solution validity.

Vedi

Page 3: Complex Absolute Value Equations

This page delves into more advanced absolute value equations, including those with multiple absolute value terms.

Key topics:

Solving equations of the form |A(x)| = k for different values of k
Handling equations with nested absolute values
Techniques for equations with multiple absolute value terms

Example: The equation |x - 1| - 3|x| = -2 is solved step-by-step, demonstrating how to handle multiple absolute value terms.

Highlight: When solving complex absolute value equations, it's crucial to consider all possible combinations of positive and negative cases for each absolute value term.

The page provides a systematic approach to tackling these challenging problems, emphasizing the importance of careful case analysis and algebraic manipulation.

Contenuti simili

Know Equazioni e disequazioni esponenziali thumbnail

2576

4ªl

Equazioni e disequazioni esponenziali

Appunti presi in classe e vari esempi sulle equazioni e disequazioni esponenziali con grafici + una piccola ripetizione sulle proprietà delle potenze

383

4ªl

Numeri complessi

numeri complessi, coordinate polari

Know Scomposizione di polinomi thumbnail

466

4ªl

Scomposizione di polinomi

Metodi, teoria e passaggi.

3009

4ªl

i Logaritmi

appunti sui logaritmi e le loro 3 proprietà

1815

4ªl

funzioni

funzioni: dominio, intersezione con gli assi, pari e dispari

Know Radicali: campo di esistenza e segno thumbnail

1985

2ªl/3ªl

Radicali: campo di esistenza e segno

Come definire il dominio e il segno dei radicali analizzando i diversi passaggi

Non c'è niente di adatto? Esplorare altre aree tematiche.

Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

Scarica

Google Play

Scarica

App Store

Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

4.9+

Valutazione media dell'app

13 M

Studenti che usano Knowunity

#1

Nelle classifiche delle app per l'istruzione in 12 Paesi

950 K+

Studenti che hanno caricato appunti

Non siete ancora sicuri? Guarda cosa dicono gli altri studenti...

Utente iOS

Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS

Adoro questa app ❤️, la uso praticamente sempre quando studio.

Materie

Matematica

Relazioni e funzioni

Funzioni goniometriche

Formule addizione sottrazione, duplicazione, bisezione

Scopri le Equazioni e Disequazioni con Valore Assoluto: Esercizi e Schemi Facili

Lele☁️🥥

@walkerele

371 Follower

Segui

Punti chiave:

Definizione e proprietà dei valori assoluti
Metodi di risoluzione per equazioni con uno o più valori assoluti
Interpretazione grafica delle equazioni con valori assoluti
Tecniche per risolvere disequazioni con valori assoluti
Approccio alle equazioni irrazionali

18/1/2023

9331

2ªl/3ªl

Matematica

298

Page 4: Synthetic Equations and General Strategies

This page covers synthetic equations involving absolute values and presents general strategies for solving absolute value equations.

Key points:

Handling equations where the absolute value term is set equal to a quadratic or other non-linear expression
General principles for solving |A(x)| = K and |A(x)| = |B(x)| type equations
A comprehensive summary schema for approaching absolute value equations

Example: The equation |x² - 9| + |x - 3| = 0 is solved, illustrating how to handle the sum of two absolute value terms.

Highlight: The general strategy involves studying the sign of expressions inside absolute value symbols and considering different cases based on possible combinations.

The page concludes with a useful summary schema that outlines the steps for solving various types of absolute value equations, providing a valuable reference for students.

Page 1: Equations with Absolute Values

This page introduces key concepts and formulas for working with absolute value equations.

Key points:

The absolute value of a number is its distance from zero on a number line
Absolute value equations often require considering multiple cases
Common absolute value identities and properties are presented

Definition: The absolute value of x, denoted |x|, is defined as x if x ≥ 0 and -x if x < 0.

Example: |x + 4| = |x| + 4 demonstrates how absolute values interact with addition.

Highlight: When solving |A(x)| = B(x), consider the cases A(x) = B(x) and A(x) = -B(x) separately.

The page provides several solved examples of absolute value equations, illustrating different solution techniques and potential pitfalls to avoid.

Page 6: Advanced Absolute Value Inequalities

This page delves deeper into absolute value inequalities, covering more complex scenarios and general solution strategies.

Key topics:

General forms of absolute value inequalities and their solutions
Handling inequalities with multiple absolute value terms
Strategies for solving |A(x)| < |B(x)| type inequalities

Definition: For K > 0, |x| < K is equivalent to -K < x < K, while |x| > K is equivalent to x < -K or x > K.

Example: The inequality |x² - 4| < 5 is solved, demonstrating how to handle quadratic expressions within absolute value symbols.

The page emphasizes the importance of understanding the behavior of absolute value functions and how this knowledge can be applied to solve complex inequalities efficiently.

Page 5: Inequalities with Absolute Values

This page introduces inequalities involving absolute values, a crucial topic in advanced algebra.

Key concepts covered:

Solving inequalities of the form |x - a| < b and |x - a| > b
Handling compound inequalities with absolute value terms
Graphical interpretation of absolute value inequalities

Example: The inequality |5x - 1| < 3x + 2 is solved step-by-step, demonstrating how to handle cases and combine solution intervals.

Highlight: When solving absolute value inequalities, it's essential to consider the direction of the inequality and how it affects the solution process.

The page provides multiple solved examples, each illustrating different techniques and potential complexities that can arise in absolute value inequalities.

Page 7: Irrational Equations

This page introduces irrational equations, focusing on equations involving square roots.

Key concepts:

Solving equations of the form √A(x) = B(x)
Importance of checking domain restrictions and extraneous solutions
Handling equations with higher index roots

Highlight: When solving irrational equations, it's crucial to check for extraneous solutions by substituting results back into the original equation.

Example: The equation √3x + 4 = 2 + x is solved step-by-step, demonstrating the process of squaring both sides and checking solutions.

The page also covers more complex irrational equations, including those with cube roots and nested radicals, providing a comprehensive overview of solution techniques.

Page 2: Graphical Interpretation of Absolute Value Equations

This page explores the graphical representation of absolute value equations and inequalities.

Key concepts covered:

Visualizing absolute value functions on a coordinate plane
Using graphs to understand solution behavior
Interpreting intersection points and regions

Example: The equation |x - 5| = 3x - 1 is analyzed graphically, showing how the solution x = 3/2 corresponds to the intersection point of two functions.

The page also covers more complex equations like |3x - 5| = 2x + 1, demonstrating how to handle piecewise functions and multiple solution intervals.

Highlight: Graphical analysis can provide valuable insights into the number and nature of solutions for absolute value equations and inequalities.

Page 8: Advanced Irrational Equations

This page covers more advanced topics in irrational equations, including systems with multiple radicals and irrational inequalities.

Key topics:

Solving equations with multiple radical terms
Techniques for handling irrational inequalities
Graphical interpretation of irrational equations and inequalities

Example: The equation 2√(2+x) - √(x-3) = 4 is solved, demonstrating how to handle equations with multiple radical terms.

Highlight: When solving irrational inequalities, it's important to consider domain restrictions and the behavior of radical functions.

The page concludes with strategies for solving complex irrational equations and inequalities, emphasizing the importance of systematic approaches and careful analysis of solution validity.

Page 3: Complex Absolute Value Equations

This page delves into more advanced absolute value equations, including those with multiple absolute value terms.

Key topics:

Solving equations of the form |A(x)| = k for different values of k
Handling equations with nested absolute values
Techniques for equations with multiple absolute value terms

Example: The equation |x - 1| - 3|x| = -2 is solved step-by-step, demonstrating how to handle multiple absolute value terms.

Highlight: When solving complex absolute value equations, it's crucial to consider all possible combinations of positive and negative cases for each absolute value term.

The page provides a systematic approach to tackling these challenging problems, emphasizing the importance of careful case analysis and algebraic manipulation.

Contenuti simili

Matematica - Equazioni e disequazioni esponenziali

Appunti presi in classe e vari esempi sulle equazioni e disequazioni esponenziali con grafici + una piccola ripetizione sulle proprietà delle potenze

2576

Matematica - Numeri complessi

numeri complessi, coordinate polari

383

Matematica - Scomposizione di polinomi

Metodi, teoria e passaggi.

466

Matematica - i Logaritmi

appunti sui logaritmi e le loro 3 proprietà

3009

Matematica - funzioni

funzioni: dominio, intersezione con gli assi, pari e dispari

1815

Matematica - Radicali: campo di esistenza e segno

Come definire il dominio e il segno dei radicali analizzando i diversi passaggi

1985

Non c'è niente di adatto? Esplorare altre aree tematiche.

Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

Scarica

Google Play

Scarica

App Store

4.9+

Valutazione media dell'app

13 M

Studenti che usano Knowunity

#1

Nelle classifiche delle app per l'istruzione in 12 Paesi

950 K+

Studenti che hanno caricato appunti

Non siete ancora sicuri? Guarda cosa dicono gli altri studenti...

Utente iOS

Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS