≤ (Less Than or Equal To)

Definition

The symbol ≤ represents 'less than or equal to' and is used to indicate that a value is smaller than or equal to another value. It is a relational operator that compares two quantities and determines their relative size or magnitude. This term is crucial in the context of solving linear inequalities, graphing linear inequalities, and working with systems of linear inequalities.

5 Must Know Facts For Your Next Test

1. The symbol ≤ indicates that a value is less than or equal to another value, whereas the symbol < indicates that a value is strictly less than another value.
2. When solving linear inequalities, the ≤ symbol is used to represent the boundary of the solution set, which includes the values that make the inequality true.
3. In the context of graphing linear inequalities, the ≤ symbol is used to determine the shaded region that represents the solution set on the coordinate plane.
4. When working with systems of linear inequalities, the ≤ symbol is used to define the constraints that the solution must satisfy, and the feasible region is the intersection of the individual solution sets.
5. The ≤ symbol is also used in applications involving linear inequalities, such as in optimization problems where the goal is to find the maximum or minimum value subject to certain constraints.

Review Questions

• Explain how the ≤ symbol is used in the context of solving linear inequalities.
  • When solving linear inequalities, the ≤ symbol is used to represent the boundary of the solution set. This means that the values that make the inequality true, including the value at the boundary, are considered part of the solution set. For example, the inequality $x ≤ 5$ has a solution set that includes all real numbers less than or equal to 5, such as -2, 0, 3.5, and 5.
• Describe the role of the ≤ symbol in the context of graphing linear inequalities.
  • In the context of graphing linear inequalities, the ≤ symbol is used to determine the shaded region that represents the solution set on the coordinate plane. The shaded region includes all the points that satisfy the inequality, with the boundary line being part of the solution set. For example, the graph of the inequality $y ≤ 3x + 2$ would be a half-plane, with the boundary line $y = 3x + 2$ included in the solution set.
• Analyze how the ≤ symbol is used in the context of systems of linear inequalities and their applications.
  • When working with systems of linear inequalities, the ≤ symbol is used to define the constraints that the solution must satisfy. The feasible region, which represents the set of all possible solutions, is the intersection of the individual solution sets of the linear inequalities in the system. In applications involving linear inequalities, such as optimization problems, the ≤ symbol is used to represent the constraints that must be met in order to find the maximum or minimum value of the objective function.

Related terms

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

Linear Inequality: A linear inequality is an inequality that involves a linear expression, such as $ax + b ≤ c$, where $a$, $b$, and $c$ are constants.

System of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities that must be satisfied simultaneously.