INECUACIONES DE 1° GRADO

Introduction to First-Degree Inequalities

Overview of First-Degree Inequalities

• The instructor, Licenciado Bolívar, introduces the topic of first-degree inequalities in the form AX > B, AX < B, AX ≥ B, and AX ≤ B, emphasizing that A must be non-zero.

• It is noted that the solution set for these inequalities consists of infinite intervals; thus, understanding how to read and represent these intervals is crucial.

Understanding Interval Representation

• The instructor explains how to visualize inequalities using hand gestures: for example, "x > a" extends forward from point 'a'.

• The distinction between open intervals (for > or <) and closed intervals (for ≥ or ≤) is highlighted with graphical representations.

Direction of Inequalities

• When considering "x < b", the representation points backward from 'b', indicating values less than 'b'.

• Similar reasoning applies for "x ≤ b", reinforcing the importance of interval notation in representing solutions.

Key Theorem for Solving Inequalities

Consequence of Order Axioms

• A theorem states that if A < B and C is a negative number, multiplying both sides by C reverses the inequality: A * C > B * C.

• This principle applies universally across all types of inequalities (>, <, ≥, ≤), stressing that multiplication by a negative changes the direction of the inequality.

Coefficient Considerations

• For solving inequalities correctly, it’s essential that variables have positive coefficients. If not, adjustments must be made to maintain accuracy in solutions.

Examples and Applications

Example 1: Analyzing Solutions

• The first example involves finding solutions for three given inequalities. It discusses scenarios where an inequality may yield either a true or false result.

• In this case study, if an inequality results as false (e.g., 0 > 3), then the solution set is empty; conversely, if true (all real numbers), it encompasses all possible values.

Example 2: Distributing Terms

• In another example involving distribution across terms like 1/4 - 3y , careful manipulation leads to determining whether y leq 0 .

• Here it’s clarified that while 0y < 0 might be false under strict conditions, equality allows for specific solutions such as y = c .

Example 3: Further Distribution Challenges

• The final example illustrates distributing terms on both sides leading to simplified expressions.

• Ultimately concluding with valid statements about variable relationships based on derived inequalities.

Understanding Inequalities and Their Solutions

Key Concepts in Solving Inequalities

• The solution to inequalities involves all real numbers, often resulting in infinite intervals. Care must be taken with the coefficient of the variable, which should be positive for proper manipulation.

• The process of solving inequalities mirrors that of equations: addition becomes subtraction, subtraction becomes addition, multiplication becomes division, and vice versa. A critical point is that multiplying or dividing by a negative number reverses the inequality.

Step-by-Step Example 1

• Starting with the inequality 2(3x - 5) - x geq 1 - 2x - 3, distribute terms to simplify it into a more manageable form. This leads to 6x - 10 - x geq 1 + 2x + 3.

• Rearranging terms separates variables from constants while ensuring the variable's coefficient remains positive. This results in an expression where 7x is isolated on one side.

• The final result shows x geq 2. On a number line, this is represented as a closed interval starting at 2 and extending to infinity.

Solution Representation

• The solution can also be expressed in set notation: x in mathbbR : x geq 2. This indicates that all values greater than or equal to two are included in the solution set.

Step-by-Step Example 2

• For the second example involving 2 - 4y - 1/3 > 3 - y, begin by simplifying each term appropriately. Multiplying through by a positive denominator maintains the direction of the inequality.

• When coefficients are negative, such as when multiplying by -1, remember that this will reverse the inequality sign. Thus, if you start with something like >2, it changes to < -2.

Final Solution Interpretation