Formas de representar la solución de una Desigualdad.

Understanding Inequalities: Graphical and Interval Representations

Introduction to Inequalities

• The video begins with an introduction by Profe Javier, focusing on how to represent the solution of an inequality.

• The first example discussed is the inequality x > 3 , which can also be expressed as x geq 3 .

Algebraic Representation

• The algebraic solution involves isolating x . For instance, if we have x > 3 , it indicates that x can take any value greater than 3.

• Values such as 4, 5, and even larger numbers up to infinity are included in this set. The left endpoint is defined at 3, while the right extends towards positive infinity.

Graphical Representation

• To graphically represent this inequality on a number line:

• A point is marked at 3 (the left endpoint).

• An arrow extends from this point towards positive infinity to indicate all values greater than 3.

• It’s crucial to denote whether the endpoint (in this case, 3) is included in the solution. Since it’s not included in x > 3 , a small circle is drawn around it.

Second Example: Inclusive Inequality

• In contrast, for the inequality x geq 3 :

• The number line again starts with negative infinity on the left and positive infinity on the right.

• Here, since we include the value of 3, it will be represented differently compared to when it's exclusive.

Inclusion vs. Exclusion in Graphing

• If an equality sign accompanies an inequality (like in x geq 3 ), then that endpoint (here, 3) must be highlighted or filled in on the graph.

• Conversely, if there’s no equality sign (as with x > 3 ), then that endpoint remains unfilled.

Interval Notation

• Transitioning from graphical representation to interval notation:

• For x > 3, it translates into interval notation as (3, +infty) .

• Parentheses are used because neither endpoint is included; thus both endpoints remain open.

Final Example: Mixed Inclusion

• Another example reiterates these concepts using different inequalities:

• For intervals where one side includes its boundary (like with inclusive inequalities), square brackets are used instead of parentheses.

• This reinforces understanding of how inclusion affects both graphical and interval representations.

By following these structured notes based on timestamps from Profe Javier's video lecture, learners can effectively grasp how inequalities are represented both algebraically and graphically.