key term - Open Circle

Definition

An open circle is a mathematical symbol used to represent a strict inequality in the context of linear and rational inequalities. It denotes that the solution set does not include the boundary point, indicating that the inequality is strictly less than or strictly greater than the given value.

5 Must Know Facts For Your Next Test

1. An open circle is used to represent a strict inequality, where the solution set does not include the boundary point.
2. In the context of linear inequalities, an open circle is used to indicate that the solution set is strictly less than or strictly greater than the given value.
3. When solving rational inequalities, an open circle is used to represent that the solution set does not include the value where the denominator is equal to zero.
4. The open circle symbol is typically used in conjunction with other inequality symbols, such as less than (<) or greater than (>).
5. The open circle symbol is visually distinct from the closed circle symbol, which represents a non-strict inequality where the solution set includes the boundary point.

Review Questions

• Explain the role of the open circle symbol in the context of solving linear inequalities.
  • In the context of solving linear inequalities, the open circle symbol is used to indicate that the solution set does not include the boundary point. For example, the inequality $x < 5$ would be represented with an open circle at the point $x = 5$, signifying that the solution set includes all values strictly less than 5, but not the value 5 itself. This is in contrast to the closed circle symbol, which would be used for the non-strict inequality $x \leq 5$, where the solution set includes the boundary point of 5.
• Describe how the open circle symbol is used when solving rational inequalities.
  • When solving rational inequalities, the open circle symbol is used to represent the values where the denominator of the rational expression is equal to zero. These values are excluded from the solution set, as they would result in an undefined or infinite value. For example, the inequality $\frac{x - 3}{x - 1} > 0$ would have an open circle at $x = 1$, since this is the value where the denominator is equal to zero. The solution set would include all values of $x$ strictly greater than 1, but not the value 1 itself.
• Analyze the differences between the open circle and closed circle symbols in the context of inequality solutions.
  • The open circle and closed circle symbols represent fundamentally different types of inequalities and their corresponding solution sets. The open circle symbol denotes a strict inequality, where the boundary point is excluded from the solution set. In contrast, the closed circle symbol represents a non-strict inequality, where the boundary point is included in the solution set. This distinction is crucial when solving both linear and rational inequalities, as it determines whether the values at the boundary are considered part of the final solution or not. Understanding the appropriate use of these symbols and their implications is essential for accurately representing and solving various types of inequalities.

Related terms

Closed Circle:

A closed circle is a mathematical symbol used to represent a non-strict inequality, where the solution set includes the boundary point.

Inequality:

An inequality is a mathematical statement that compares two values, using symbols such as less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥).

Solution Set:

The solution set of an inequality is the set of all values that satisfy the given inequality.