Desigualdades dobles. Ejemplo 2.

Understanding Double Inequalities with Fractional Terms

Transforming Fractional Inequalities to Integer Form

• The video begins by addressing the transformation of fractional inequalities into integer inequalities by multiplying through by the least common multiple (LCM) of the denominators. In this case, the LCM is 5.

• The instructor emphasizes that since they are multiplying by a positive number (5), the direction of the inequality remains unchanged throughout the process. This is crucial for maintaining the integrity of the inequality.

• After multiplication, each term in the inequality is simplified: 5x - 4/5 becomes x - 2 , and -5/5 simplifies to -1, leading to an integer form of the inequality.

Solving for x in Integer Form

• The next step involves isolating x . The instructor subtracts 2 from both sides, keeping in mind that this does not change the direction of the inequality as it’s a simple subtraction operation. Thus, we have -3x < -1 + 2 .

• As part of solving for x , dividing both sides by -3 requires flipping the inequality sign due to division by a negative number, resulting in x > 22/3 . This critical point highlights how operations affect inequalities differently based on their nature (positive or negative).

Graphical Representation on a Number Line

• Transitioning to graphical representation, a number line is introduced where key points such as 1/3 and 22/3 are marked clearly. This visual aid helps illustrate where solutions lie relative to these values on a continuum.

• The instructor explains how to interpret these points: values greater than or equal to 1/3 are shaded towards infinity on one side while values less than but not including 22/3 extend towards negative infinity on another side. This dual shading indicates valid solution ranges visually.

Final Solution Interpretation

• A final review clarifies that only sections fulfilling both conditions—greater than or equal to 1/3 and less than 22/3 —are considered valid solutions within specified intervals on the number line. Thus, it reinforces understanding through visual confirmation alongside algebraic manipulation.

• The conclusion reiterates that closed intervals indicate included endpoints while open intervals signify excluded ones; hence proper notation is essential for conveying complete information about solution sets effectively.