Analyzing Systems with Parameters

This page delves deeper into solving systems of inequalities that involve parameters. It demonstrates how the solution can vary depending on the value of the parameter and introduces a systematic approach to analyzing such systems.

The main example discussed is the inequality x ≤ 5a, where a is a parameter. The page outlines three cases to consider:

1. When 5a < 2
2. When 5a = 2
3. When 5a > 2

Example: For the case 5a < 2, the solution is derived as a < 2/5.

The page introduces a general schema for analyzing inequalities with parameters:

1. Equazioni Intere (Whole Equations)
2. Disequazioni Intere (Whole Inequalities)
3. Equazioni Fratte (Fractional Equations)
4. Disequazioni Fratte (Fractional Inequalities)

Highlight: This schema provides a structured approach to solving complex systems of inequalities, especially those involving parameters.

The concept of "forma normale" (normal form) is introduced, which is crucial for standardizing inequalities for easier comparison and solution.

Definition: The normal form of an inequality is a standard way of writing the inequality, typically with the variable terms on one side and the constant terms on the other.

The page concludes with a discussion on special cases in systems of inequalities, such as when a system is always true (sempre) or impossible (impossibile).

Vocabulary: "Sempre" means "always" in Italian, indicating a system that is true for all values of the variable. "Impossibile" means "impossible", referring to a system that has no solution.

This comprehensive guide provides students with the tools to tackle complex systems of inequalities, including those with parameters, and understand the nuances of their solutions.

Understanding Systems of Inequalities

This page introduces the concept of systems of inequalities and provides detailed examples of solving various types of inequality systems. The focus is on understanding how to find values of x that simultaneously satisfy multiple inequalities.

Definition: A system of inequalities is a set of inequalities that must be solved simultaneously, finding the values of the variable that satisfy all inequalities in the system.

The page presents several examples to illustrate different types of systems:

1. A system with three inequalities:
   • A: -x ≤ 0
   • B: 2(x+1) ≤ x+1
   • C: 1-(x+1) ≤ -2

Example: For the inequality B: 2(x+1) ≤ x+1, the solution process is shown step-by-step, resulting in x ≤ -1.

2. A system involving fractional inequalities: x(1/2 - 1/3) ≤ (x-1)(1/3 - 1/4)

Highlight: The solution to this fractional inequality is derived as x ≤ 2.

3. A system with a parameter: 3x - 3a ≤ 2x + 2a, which is simplified to x ≤ 5a

Vocabulary: A parameter is a variable whose value is not specified in the inequality but can affect the solution.

The page also emphasizes the importance of graphical representations in solving systems of inequalities. It shows how to visualize the solution set on a number line, which is particularly useful for understanding the intersection of multiple inequalities.

Highlight: The final solution to a system of inequalities is often represented as an interval or a union of intervals on the real number line.