<

Definition

The symbol '<' represents a mathematical inequality indicating that the value on the left is less than the value on the right. This symbol is used to compare two quantities and helps in formulating linear inequalities, which are essential in both real-world applications and graphical representations of relationships between variables.

5 Must Know Facts For Your Next Test

1. < indicates that one number is strictly less than another, which is key in forming linear inequalities and understanding their solutions.
2. When graphing linear inequalities, the area representing solutions is typically shaded to indicate all the values that satisfy the inequality.
3. In a one-variable context, solving an inequality involving < can involve isolating the variable just like with equations, but the solution will be expressed as a range.
4. For systems of linear inequalities, multiple inequalities can intersect, and the solution set will be where all shaded regions overlap on the graph.
5. When multiplying or dividing both sides of an inequality by a negative number, it is important to reverse the inequality sign; for example, if you have -x < 4 and divide by -1, it becomes x > -4.

Review Questions

• How do you solve a linear inequality involving the symbol <, and what does this indicate about the possible values of the variable?
  • To solve a linear inequality with '<', you isolate the variable on one side just like in an equation. For example, if you start with 2x + 3 < 7, you would subtract 3 from both sides to get 2x < 4, then divide by 2 to find x < 2. This indicates that any value less than 2 will satisfy the inequality, meaning x can take on an infinite number of values up to but not including 2.
• What are the graphical implications of using < in a system of linear inequalities when representing multiple inequalities on a coordinate plane?
  • In a system of linear inequalities where one or more inequalities use '<', the resulting graph will consist of shaded regions indicating where all solutions exist. The boundary line for an inequality using '<' is typically dashed, showing that points on this line are not included in the solution set. The overlapping shaded areas represent values satisfying all inequalities in the system simultaneously.
• Evaluate how understanding the symbol < contributes to decision-making in real-world applications, especially in scenarios involving constraints.
  • Understanding '<' allows individuals to define limits and constraints in real-world situations such as budgeting, resource allocation, or risk assessment. For example, if a company cannot spend more than $5000 on marketing (represented as spending < $5000), it helps decision-makers set realistic goals and evaluate options within financial limits. By applying these inequalities to model scenarios mathematically, businesses can optimize their strategies while adhering to necessary constraints.

Related terms

Inequality: A mathematical statement that shows the relationship between two expressions that are not equal, often using symbols such as <, >, ≤, or ≥.

Boundary Line: A line that represents the solution set of an inequality in a graph, dividing the coordinate plane into regions where the inequality holds true.

Solution Set: The set of all possible values that satisfy a given inequality or system of inequalities.