Desigualdades con Valor Absoluto. Video 2

Understanding Absolute Value Inequalities

Introduction to Absolute Value Properties

• The instructor, Profe Gabriel, introduces the topic of absolute value inequalities and emphasizes the importance of noting key properties related to absolute values.

Case 2: Setting Up the Inequality

• The exercise presented involves the algebraic expression |x/3 + 1| leq 4, which corresponds to Case 2 where an absolute value is less than or equal to a constant b.

• The solution approach requires setting up a double inequality: -b leq textexpression leq b.

Transforming the Expression

• The negative extreme is set as -4, leading to the transformation of the original expression into a double inequality format.

• To eliminate fractions, the entire inequality is multiplied by 3 (the denominator), simplifying it into an integer form.

Solving for x

• After multiplying through by 3, the resulting inequality becomes -12 leq x leq 9.

• Rearranging gives us two conditions: x geq -15 and x leq 9.

Graphical Representation of Solutions

• A number line is used to graphically represent solutions with points at -15 (to the left of zero) and 9 (to the right).

• The closed interval notation indicates that both endpoints are included in the solution set.

Final Solution Interpretation

• The final solution states that x must be between -15 and 9, inclusive. This means all values within this range satisfy the original inequality.

• The valid interval for both conditions combined is expressed as [-15, 9], indicating a closed interval on both ends.

This structured summary provides a clear understanding of how to solve absolute value inequalities using specific cases and graphical representation techniques.