Question

Use inequality notation to describe the subset of real numbers.\(m\) is at least \(-5\) and at most \(9 .\)

Short answer

The inequality representation of the subset of real numbers where \(m\) is at least -5 and at most 9 is \(-5 \leq m \leq 9\).

Step-by-step solution

Understanding the value of \(m\)

The exercise says that \(m\) is at least -5. 'At least' in mathematics means 'greater than or equal to'. So, this part gives us the inequality \(m \geq -5\). This means that the value of \(m\) is -5 or any number larger than -5.

The upper limit of \(m\)

The next part of the exercise says that \(m\) is at most 9. 'At most' in mathematics means 'less than or equal to'. Therefore, this part gives us the inequality \(m \leq 9\). It means that the value of \(m\) is 9 or any number less than 9.

Combine the inequalities

Taking both step 1 and step 2 into account, we can say that -5 is less or equal to \(m\) and \(m\) is less or equal to 9. This can also be written as \(-5 \leq m \leq 9\). This inequality equation represents a range of numbers that includes all real numbers between and including -5 and 9.

Key concepts

Understanding Real Numbers

Real numbers are the set of numbers that include all the numbers on the number line, which means they consist of both rational and irrational numbers.

For example, \textbf{rational numbers} like \(\frac{1}{2}\), \(3\), and \(-8\), as well as \textbf{irrational numbers} like \(\text{\(\text{π}\text{\)}}\) and \(\text{\(\text{√2}\text{\)}}\), are all considered real numbers. They can be positive, negative, or zero. When we deal with inequality notation in mathematics, as in the given exercise, we are often describing a subset of these real numbers.

The importance of understanding real numbers in this context lies in being able to identify the possible values that a variable, such as \(m\) from our exercise, can take.

Decoding Mathematical Inequalities

Mathematical inequalities are statements that relate two expressions with an inequality sign. The basic inequality signs include

• \textbf{Greater than (\(>\text{\)}})
• \textbf{Less than (\(<\)}): Indicates that a value on the left is smaller than the value on the right.
• \textbf{Greater than or equal to (\(\geq\)}): As we saw with \(m \text{\}\geq -5\), this means the value can be equal to or greater than the other.
• \textbf{Less than or equal to (\(\leq\)}): Mirroring the previous, \(m \leq 9\) informs us that \(m\) can be equal to or less than 9.

By combining inequalities, we create a range that specifies the possible values for a variable. In this case, the range is from \(-5\) to \(9\), including both endpoints, indicating a closed interval on a number line. Inequalities are essential for describing intervals, expressing constraints, and defining domain and range in functions.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can represent a range of values and are fundamental in forming equations and inequalities.

In the context of our exercise, \(m\) is an algebraic expression that represents an unknown value within the constraints of the inequalities \(-5\) and \(9\). Algebraic expressions become especially powerful when we combine them with inequality notation to describe a set of possible numbers.

Understanding how to read and write these expressions is crucial in algebra and other areas of mathematics because it allows us to express relationships and solve problems involving unknown variables.