Question

Write a verbal phrase that describes the inequality. \(9 \leq x\)

Short answer

The verbal phrase for the inequality \(9 \leq x\) is 'The variable \(x\) is at least 9 or greater than 9'.

Step-by-step solution

Identify the inequality symbol

First, recognize the inequality symbol which is \(\leq\). This symbol stands for 'less than or equal to'.

Read the inequality with symbol

Next, read the inequality along with symbol. We have \(9 \leq x\), it means the number 9 is less than or equal to the variable \(x\).

Formulate the verbal phrase

From the above, the verbal phrase that describes the inequality is 'The variable \(x\) is at least 9 or greater than 9'

Key concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions comparing two values, quantities, or expressions with the use of inequality symbols. Unlike equations, which show equality, inequalities express a relation where one side is not necessarily equal to the other but may also be greater or smaller.

Understanding algebraic inequalities is crucial for solving real-world problems involving limits, constraints, or conditions. For instance, determining whether a budget is sufficient for a list of expenses, or figuring out the possible dimensions of a parcel that need to fit within a predefined space are all situations where inequalities are applicable.

To solve an inequality is to find all values of the variable that make the inequality true. These values are collectively referred to as the solution set. For the inequality provided, \(9 \leq x\), any number greater than or equal to 9 would be part of the solution set.

Inequality Symbols

Inequality symbols are the notations that represent the comparative value between two entities. These notations are foundational for conveying the relationship in a concise mathematical form. The primary inequality symbols include:

• \(<\) for 'less than'
• \(>\) for 'greater than'
• \(\leq\) for 'less than or equal to'
• \(\geq\) for 'greater than or equal to'

The symbol \(\leq\), as used in the given exercise, is particularly important as it includes the possibility of equality, indicating that the value on the left side is either less than or exactly equal to the value on the right side. In our example, the number 9 is either less than or equal to the variable \(x\).

Verbal Representation of Inequalities

Translating algebraic inequalities into verbal phrases aids in better comprehension and allows individuals to communicate mathematical concepts without the need for symbols. The verbal representations of inequalities usually follow standard phrases that correlate with the symbols.

For the inequality \(9 \leq x\), the verbal phrase is 'The variable \(x\) is at least 9 or is greater than 9.' This phrase captures the essence of the inequality symbol \(\leq\) by indicating that \(x\) cannot be less than 9 but could be 9 or any number larger.

Verbal phrases are not only useful in academics but also in everyday scenarios that involve variables and constraints, allowing for a clearer understanding and a better articulation of mathematical relationships in spoken language.