key term - ≤ (Less Than or Equal To)

Definition

The symbol '≤' represents the mathematical concept of 'less than or equal to.' It is used to indicate that a value or quantity is less than or equal to another value or quantity. This term is commonly used in the context of inequalities, where it helps define the range of values that satisfy a given condition. The understanding of the symbol '≤' is crucial in the study of linear inequalities, absolute value inequalities, and systems of nonlinear equations and inequalities involving two variables.

5 Must Know Facts For Your Next Test

1. The symbol '≤' is used to indicate that a value or quantity is less than or equal to another value or quantity.
2. In the context of linear inequalities, the symbol '≤' is used to represent the set of solutions where the left-hand side of the inequality is less than or equal to the right-hand side.
3. Absolute value inequalities involving '≤' represent the set of values for the variable where the distance between the variable and a given value is less than or equal to a specified amount.
4. When solving systems of nonlinear equations and inequalities with two variables, the symbol '≤' is used to define the feasible region, which represents the set of points that satisfy all the constraints in the system.
5. The understanding of the symbol '≤' and its implications is crucial for solving a wide range of problems in college algebra, including optimization, graphing, and decision-making.

Review Questions

• Explain how the symbol '≤' is used in the context of linear inequalities.
  • In the context of linear inequalities, the symbol '≤' is used to represent the set of solutions where the left-hand side of the inequality is less than or equal to the right-hand side. For example, the inequality $2x + 3 ≤ 7$ describes the set of values for $x$ where the expression $2x + 3$ is less than or equal to $7$. This allows for the identification of the range of values that satisfy the given condition, which is essential in solving problems involving linear inequalities.
• Describe how the symbol '≤' is used in the context of absolute value inequalities.
  • In the context of absolute value inequalities, the symbol '≤' is used to represent the set of values for the variable where the distance between the variable and a given value is less than or equal to a specified amount. For instance, the inequality $|x - 3| ≤ 5$ describes the set of values for $x$ where the distance between $x$ and $3$ is less than or equal to $5$. This allows for the identification of the range of values that satisfy the given condition, which is crucial in solving problems involving absolute value inequalities.
• Explain the role of the symbol '≤' in the context of systems of nonlinear equations and inequalities with two variables.
  • When solving systems of nonlinear equations and inequalities with two variables, the symbol '≤' is used to define the feasible region, which represents the set of points that satisfy all the constraints in the system. The feasible region is the area where the solutions to the system of equations and inequalities intersect, and it is bounded by the constraints, including those involving the '≤' symbol. Understanding the implications of the '≤' symbol in this context is essential for solving optimization problems and making informed decisions based on the feasible region.

Related terms

Inequality:

A mathematical statement that compares two expressions using comparison symbols like ≤, <, ≥, >, or ≠, indicating that the values are not equal.

Linear Inequality:

An inequality that can be written in the form $ax + by ≤ c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables.

Absolute Value Inequality:

An inequality that involves the absolute value of an expression, such as $|x - 3| ≤ 5$, which represents the set of values for $x$ where the distance between $x$ and $3$ is less than or equal to $5$.