Interval notation is a way to represent a range of numbers or values using a specific set of symbols and conventions. It is commonly used to describe the solutions or solutions sets of various types of inequalities, as well as to graph and visualize these solutions on a number line.
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Interval notation uses square brackets '[ ]' to represent closed intervals (including the endpoints) and parentheses '( )' to represent open intervals (excluding the endpoints).
Interval notation can be used to represent the solutions of linear, compound, absolute value, and quadratic inequalities.
When graphing systems of linear inequalities, interval notation is used to describe the feasible region, which is the area where all the inequalities are satisfied simultaneously.
In the context of solving quadratic inequalities, interval notation is used to represent the range of values for the variable that satisfy the inequality.
Interval notation is a concise and efficient way to communicate the solutions of various types of inequalities, making it a crucial skill for solving and understanding these mathematical concepts.
Review Questions
Explain how interval notation is used to represent the solutions of linear inequalities.
When solving linear inequalities, interval notation is used to represent the range of values that satisfy the inequality. For example, the solution to the inequality $x > 3$ would be written in interval notation as '(3, \infty)', indicating that all values greater than 3 are part of the solution set. Similarly, the solution to $x \leq -2$ would be written as '(-\infty, -2]', representing all values less than or equal to -2.
Describe how interval notation is used in the context of solving compound inequalities.
Compound inequalities involve the use of the 'and' (conjunction) or 'or' (disjunction) operators to combine two or more inequalities. When solving compound inequalities, interval notation is used to represent the solution set. For a conjunction, such as '$-4 \leq x \leq 2$', the solution would be written as '[-4, 2]', indicating a closed interval that includes all values between -4 and 2, inclusive. For a disjunction, such as '$x < -1 \text{ or } x > 3$', the solution would be written as '(-\infty, -1) \cup (3, \infty)', representing the union of two open intervals.
Analyze how interval notation is utilized in the graphing of systems of linear inequalities.
When graphing a system of linear inequalities, the feasible region is the area where all the inequalities are satisfied simultaneously. Interval notation is used to describe the boundaries of this feasible region. For example, if a system of inequalities has the solution set '$x \geq 2$' and '$y \leq 4$', the feasible region would be represented in interval notation as '[2, \infty) \times (-\infty, 4]', indicating that the x-values are greater than or equal to 2, and the y-values are less than or equal to 4. This interval notation provides a concise and informative way to communicate the solutions of the system.