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Definition

The symbol '<' represents a relational operator used to indicate that one value is less than another. This operator is fundamental in understanding inequalities, helping to establish relationships between numbers and facilitating comparisons in algebraic expressions. It plays a crucial role in solving inequalities and working with absolute values, allowing for the expression of conditions and constraints in mathematical problems.

5 Must Know Facts For Your Next Test

1. The symbol '<' can be used to compare integers, fractions, decimals, and algebraic expressions, indicating which is smaller.
2. When working with inequalities, the direction of the inequality symbol can change when both sides are multiplied or divided by a negative number.
3. Inequalities involving absolute values can result in two separate inequalities when solving, leading to distinct solution sets.
4. Graphing an inequality like 'x < 5' on a number line involves shading all numbers to the left of 5 and using an open circle at 5 to indicate it is not included.
5. Understanding how to manipulate inequalities is crucial for solving complex equations and systems in algebra.

Review Questions

• How does the '<' symbol interact with other inequality symbols when solving compound inequalities?
  • When solving compound inequalities that involve the '<' symbol, it's essential to maintain the direction of each inequality based on operations performed. For instance, if you have 'x < 3' and you add 2 to both sides, the inequality remains valid: 'x + 2 < 5'. However, if you multiply both sides by a negative number, the direction of the inequality flips. Understanding these interactions ensures correct solutions for compound inequalities.
• What steps would you take to solve an absolute value inequality that includes the '<' symbol?
  • To solve an absolute value inequality like |x| < 3, you would first rewrite it as two separate inequalities: -3 < x < 3. This method captures all values of x that fall within the range defined by the absolute value condition. After setting up the two inequalities, you'd solve for x and express the solution set clearly, noting that any solution must satisfy both parts of the compound inequality.
• Analyze how changing an inequality from 'x < 7' to 'x > 7' affects the solution set and graph representation.
  • When changing an inequality from 'x < 7' to 'x > 7', the solution set changes dramatically. The first inequality represents all values less than 7, which corresponds to a shaded region on the left of 7 on a number line. In contrast, 'x > 7' encompasses all values greater than 7, shifting the shaded area to the right. This illustrates not only a numerical shift but also affects potential applications in real-world scenarios where conditions change based on thresholds.

Related terms

Inequality: A mathematical statement that compares two expressions using relational operators like '<', '>', '≤', or '≥'.

Absolute Value: The distance of a number from zero on the number line, always expressed as a non-negative value, denoted by |x|.

Solution Set: The set of all possible solutions that satisfy an inequality or equation, often represented on a number line.