Desigualdades con Valor Absoluto. Video1

Understanding Absolute Value Inequalities

Introduction to Absolute Value Inequalities

• The video begins with an introduction by Professor Gabriel, focusing on the study of inequalities involving absolute values. He emphasizes the importance of noting key properties related to absolute values.

Solving the First Example

• The first exercise presented is |4x - 6| ≥ 1. This corresponds to a specific case in the table of absolute value properties, indicating that it involves an algebraic expression compared to a determined value.

• To solve this inequality, it is transformed into two separate algebraic expressions:

• The first solution considers |4x - 6| ≤ -1 (negative case). After rearranging and simplifying, we find x ≤ 5/4 as one solution.

• When isolating x in the first part of the inequality, it remains valid since dividing by a positive number does not change the direction of the inequality symbol. Thus, x ≤ 5/4 is confirmed as a valid solution.

Finding Additional Solutions

• The second part of solving |4x - 6| ≥ 1 leads us to consider |4x - 6| ≥ 1 (positive case). Rearranging gives us another expression:

• Here we find that x ≥ 7/4 after similar steps of isolation and simplification. Again, dividing by a positive number keeps the inequality direction unchanged.

Graphical Representation

• A graphical representation on a number line is introduced next:

• The points corresponding to both solutions (5/4 and 7/4) are marked on a number line with closed circles indicating inclusive boundaries for both solutions (≤ and ≥). This visually represents where x can lie based on our findings from earlier calculations.

Interval Notation for Solutions

• Finally, Professor Gabriel explains how to express these solutions in interval notation:

• For x ≤ 5/4, this translates into (-∞, 5/4].

• For x ≥ 7/4, it becomes [7/4, +∞).

• Both intervals are combined using union notation: (-∞, 5/4] ∪ 7/4, +∞). This captures all possible values satisfying either condition derived from our original inequality problem. [