key term - ≤

Definition

The symbol '≤' is a mathematical operator that represents 'less than or equal to'. It is used to compare two values and indicates that the value on the left side is less than or equal to the value on the right side.

5 Must Know Facts For Your Next Test

1. The symbol '≤' is used to represent 'less than or equal to' in mathematical expressions and inequalities.
2. Inequalities with the '≤' symbol can be used to describe a range of values, such as $x ≤ 5$ which means all values of $x$ that are less than or equal to 5.
3. Systems of nonlinear equations and inequalities with two variables can be solved using a variety of methods, including graphing, substitution, and elimination.
4. The solution to a system of nonlinear inequalities with two variables is the intersection of the solution sets of the individual inequalities.
5. Nonlinear inequalities can be used to model real-world situations, such as optimization problems or constraints in a system.

Review Questions

• Explain how the symbol '≤' is used in the context of systems of nonlinear equations and inequalities with two variables.
  • In the context of systems of nonlinear equations and inequalities with two variables, the symbol '≤' is used to represent the 'less than or equal to' relationship between the variables. For example, an inequality like $x^2 + y^2 ≤ 25$ would describe a circular region in the coordinate plane where the sum of the squares of the $x$ and $y$ coordinates is less than or equal to 25. The solution to a system of nonlinear inequalities with the '≤' symbol would be the intersection of the solution sets of the individual inequalities.
• Describe the different methods that can be used to solve systems of nonlinear equations and inequalities with two variables that involve the '≤' symbol.
  • There are several methods that can be used to solve systems of nonlinear equations and inequalities with two variables that involve the '≤' symbol. Graphing is a common method, where the individual inequalities are graphed, and the solution set is the intersection of the regions satisfying each inequality. Substitution and elimination methods can also be used, where one variable is expressed in terms of the other and then substituted back into the other inequality. Additionally, advanced techniques like using matrices or transforming the system into a simpler form may be employed to find the solution set for systems of nonlinear inequalities with the '≤' symbol.
• Analyze how the '≤' symbol can be used to model real-world situations involving systems of nonlinear equations and inequalities with two variables.
  • The '≤' symbol in the context of systems of nonlinear equations and inequalities with two variables can be used to model a wide range of real-world situations. For example, in an optimization problem, the '≤' symbol could represent a constraint, such as a budget limit or a resource availability constraint. In engineering applications, nonlinear inequalities with the '≤' symbol could model design specifications or performance requirements. In economics, the '≤' symbol could represent supply and demand constraints or production limits. By understanding how to work with systems of nonlinear inequalities involving the '≤' symbol, students can develop the skills to model and solve complex real-world problems across various disciplines.

Related terms

Inequality: A mathematical statement that shows a relationship between two expressions where one is greater than, less than, or equal to the other.

Linear Inequality: An inequality that can be written in the form $ax + by ≤ c$, where $a$, $b$, and $c$ are constants.

Nonlinear Inequality: An inequality that cannot be written in the form of a linear inequality, often involving variables raised to a power or other nonlinear expressions.