Question

Middle-aged may be defined as being 40 or more and less than \(60 .\) Express this statement using inequalities.

Short answer

\( 40 \leq age < 60 \)

Step-by-step solution

Identify the age range

The problem states that middle-aged is defined as being 40 or more and less than 60.

Define the lower limit

The lower limit is 40, which means the age should be greater than or equal to 40. This can be written as: \( age \geq 40 \)

Define the upper limit

The upper limit is less than 60, which means the age should be less than 60. This can be written as: \( age < 60 \)

Combine the inequalities

Now combine both inequalities to form a single compound inequality: \( 40 \leq age < 60 \)

Key concepts

Inequalities

Inequalities are mathematical expressions that show the relationship of one quantity being larger or smaller than another. They are written using symbols like <, >, ≤, and ≥. When we use inequalities, we are often comparing numbers or variables to express a range of possible values.

For example, suppose we say 5 < x < 10. This double inequality means we're looking for numbers bigger than 5 but smaller than 10. Inequalities are great tools for expressing ranges, such as age groups or time periods.

In the provided exercise, we used inequalities to define the 'middle-aged' range. Here, '40 or more' translates to the inequality \(age \geq 40\), meaning the person must be at least 40 years old. Similarly, 'less than 60' translates to \(age < 60\). Combining these gives us the compound inequality \(40 \leq age < 60\). This inequality covers all ages from 40 up to, but not including, 60.

Age Range

An age range is a way of grouping people or things by their age. For instance, we might categorize people as children, teenagers, adults, or elderly, depending on their age.

In this exercise, 'middle-aged' refers to people who are at least 40 but younger than 60. This term helps us discuss and compare groups more easily. When working with age ranges in algebra, we often use inequalities to express them.

In our example, we wanted to define the middle-aged range using algebra. By identifying the lower limit (40 years) and the upper limit (60 years), we created our range. Writing this range algebraically, we combined two inequalities: one stating the age is greater than or equal to 40 (\(age \geq 40\)) and another stating the age is less than 60 (\(age < 60\)). Putting these together, we get \(40 \leq age < 60\), showing the full age span of the middle-aged group.

Algebraic Expressions

Algebraic expressions use numbers, variables, and operators to represent mathematical relationships. Understanding these expressions is crucial for solving equations and inequalities.

In our exercise, we formed the algebraic expression \(40 \leq age < 60\) to represent the age range for the middle-aged group. This expression combines two inequalities into a single statement.

The key parts of algebraic expressions include:

• **Variables:** Symbols that represent unknown values (e.g., 'age' in our case).
• **Numbers:** Specific values or constants (e.g., 40 and 60).
• **Operators:** Symbols that show operations or comparisons (e.g., ≤ and <).

In algebra, such expressions can be used to define a logical range of values. By using variables and inequalities together, we can make precise statements about ranges or intervals, just as we did with the concept of being middle-aged.