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Definition

The less-than symbol, <, is a mathematical operator used to indicate that one value is strictly smaller than another value. It is a fundamental symbol in the context of linear inequalities and absolute value inequalities, where it helps define the range of values that satisfy the given inequality.

5 Must Know Facts For Your Next Test

1. The less-than symbol, <, is used to indicate that a value is strictly smaller than another value, meaning the values cannot be equal.
2. In the context of linear inequalities, the less-than symbol is used to define the set of all values of the variable that satisfy the inequality.
3. For absolute value inequalities, the less-than symbol is used to define the range of values that are within a certain distance from a given point.
4. Compound inequalities can be formed by combining two or more less-than or greater-than symbols, creating a range of acceptable values for the variable.
5. The less-than symbol is a fundamental tool in solving and graphing both linear and absolute value inequalities, as it helps determine the solution set and the visual representation of the inequality.

Review Questions

• Explain how the less-than symbol, <, is used in the context of linear inequalities.
  • In the context of linear inequalities, the less-than symbol, <, is used to indicate that the variable must take on values that are strictly smaller than the expression on the other side of the inequality. For example, in the inequality $2x + 3 < 5x - 1$, the less-than symbol defines the set of all values of $x$ that satisfy the inequality, which means $x$ must be less than a certain value determined by the coefficients and constants in the expression.
• Describe the role of the less-than symbol in the context of absolute value inequalities.
  • When dealing with absolute value inequalities, the less-than symbol, <, is used to define the range of values that are within a certain distance from a given point. For instance, in the inequality $|x - 2| < 4$, the less-than symbol specifies that the values of $x$ must be less than 4 units away from the point $x = 2$, creating a range of acceptable values for $x$. This is because the absolute value of a number is always non-negative, so the less-than symbol is used to establish the desired distance from the given point.
• Analyze how the less-than symbol can be combined with other inequality symbols to form compound inequalities, and explain the significance of such constructions.
  • The less-than symbol, <, can be combined with other inequality symbols, such as the greater-than symbol, >, to form compound inequalities. These compound inequalities define a range of values for the variable that satisfy multiple conditions simultaneously. For example, the compound inequality $a < x < b$ indicates that the variable $x$ must be strictly greater than $a$ and strictly less than $b$, creating a bounded interval of acceptable values. Compound inequalities are particularly useful in modeling real-world situations where multiple constraints must be met, and the less-than symbol plays a crucial role in precisely defining these ranges of values.

Related terms

Linear Inequality:

An inequality that can be written in the form $ax + b < cx + d$, where $a$, $b$, $c$, and $d$ are constants, and $x$ is a variable.

Absolute Value Inequality:

An inequality that can be written in the form $|x - a| < b$, where $a$ and $b$ are constants, and $x$ is a variable.

Compound Inequality:

An inequality that involves the use of more than one inequality symbol, such as $a < x < b$, where $a$ and $b$ are constants, and $x$ is a variable.