Negative Infinity

Definition

Negative infinity is a mathematical concept that represents a value that is less than any finite number. It is denoted by the symbol '-∞' and is used to describe quantities that have no lower bound or continue to decrease without end.

5 Must Know Facts For Your Next Test

1. Negative infinity is used to represent values that are infinitely small or continue to decrease without bound.
2. When solving linear inequalities, negative infinity can be used to describe the solution set when the inequality is less than or equal to a constant.
3. The solution set for an inequality of the form $x \leq -\infty$ is the set of all real numbers less than or equal to negative infinity.
4. Negative infinity is often used in the context of graphing linear inequalities, where the graph extends indefinitely to the left, representing all values less than or equal to negative infinity.
5. Negative infinity is a theoretical concept and cannot be represented by a specific numerical value, as it is a value that is less than any finite number.

Review Questions

• Explain how negative infinity is used in the context of solving linear inequalities.
  • When solving linear inequalities, negative infinity is used to represent the solution set for inequalities that are less than or equal to a constant. For example, if the inequality is $x \leq -5$, the solution set would be all real numbers less than or equal to negative infinity, as negative infinity is less than any finite number. This means the solution set extends indefinitely to the left on the number line, representing all values that satisfy the inequality.
• Describe the relationship between negative infinity and the solution set of a linear inequality.
  • The solution set of a linear inequality that is less than or equal to a constant, such as $x \leq -5$, will include all values that are less than or equal to negative infinity. Negative infinity represents a value that is less than any finite number, so the solution set for this type of inequality will extend indefinitely to the left on the number line, encompassing all real numbers that are less than or equal to negative infinity. The solution set can be expressed as $(-\infty, -5]$, where the parenthesis and bracket indicate that negative infinity is included in the solution set.
• Analyze the role of negative infinity in the graphing of linear inequalities and explain its significance.
  • When graphing linear inequalities, negative infinity plays a crucial role in representing the solution set. For inequalities that are less than or equal to a constant, the graph will extend indefinitely to the left, representing all values that are less than or equal to negative infinity. This is because negative infinity is a mathematical concept that represents a value that is less than any finite number. The inclusion of negative infinity in the solution set ensures that the graph accurately depicts all the values that satisfy the given linear inequality. Understanding the significance of negative infinity in this context is essential for correctly interpreting and analyzing the solution set of linear inequalities.

Related terms

Infinity:

Infinity is a mathematical concept that represents a value that is greater than any finite number. It is denoted by the symbol '∞'.

Linear Inequality:

A linear inequality is an inequality that involves a linear expression, such as $ax + b \geq c$, where $a$, $b$, and $c$ are real numbers.

Solution Set:

The solution set of a linear inequality is the set of all values of the variable that satisfy the inequality.