Write the following inequalities in interval notation. Type "infinity" for $\infty$.
a) $x \leq -2$
b) $x<3.2$
c) $x \geq \frac{5}{2}$
Answer (2)
Convert x ≤ − 2 to interval notation: ( − ∞ , − 2 ]
Convert x < 3.2 to interval notation: ( − ∞ , 3.2 )
Convert x ≥ 2 5 to interval notation: [ 2 5 , ∞ )
The final answer is: a) ( − ∞ , − 2 ] , b) ( − ∞ , 3.2 ) , c) [ 2 5 , ∞ )
Explanation
Understanding the Problem We need to convert the given inequalities into interval notation. Interval notation is a way to represent a set of real numbers using intervals. We will analyze each inequality separately.
Converting the first inequality For the inequality x ≤ − 2 , this means that x can be any number less than or equal to − 2 . In interval notation, this is represented as ( − ∞ , − 2 ] . The parenthesis on the left indicates that − ∞ is not included, and the square bracket on the right indicates that − 2 is included.
Converting the second inequality For the inequality x < 3.2 , this means that x can be any number less than 3.2 . In interval notation, this is represented as ( − ∞ , 3.2 ) . The parenthesis on both sides indicates that neither − ∞ nor 3.2 are included.
Converting the third inequality For the inequality x ≥ 2 5 , this means that x can be any number greater than or equal to 2 5 . In interval notation, this is represented as [ 2 5 , ∞ ) . The square bracket on the left indicates that 2 5 is included, and the parenthesis on the right indicates that ∞ is not included.
Final Answer Therefore, the inequalities in interval notation are: a) x ≤ − 2 is ( − ∞ , − 2 ] b) x < 3.2 is ( − ∞ , 3.2 ) c) x ≥ 2 5 is [ 2 5 , ∞ )
Examples
Interval notation is used in various fields, such as calculus and real analysis, to describe intervals on the real number line. For example, when describing the domain or range of a function, interval notation provides a concise way to represent the set of possible input or output values. Understanding interval notation is crucial for solving inequalities and understanding the behavior of functions.
The inequalities in interval notation are: a) ( − ∞ , − 2 ] , b) ( − ∞ , 3.2 ) , and c) [ 2 5 , ∞ ) . Each interval represents the range of values satisfying the corresponding inequality. Understanding interval notation is essential for expressing solutions to inequalities clearly and concisely.
;
