Question

Use the diagram of distances in our solar system. (Image cannot copy) The Moon's distance \(d\) (in miles) from Earth varies from about \(220,000\) miles to about \(250,000\) miles. Write an inequality to represent this fact.

Short answer

The inequality representing the variable distance of the Moon from Earth within the range of 220,000 miles and 250,000 miles is \(220,000 \leq d \leq 250,000\).

Step-by-step solution

Understand the provided fact

Understand that the Moon's distance from Earth varies from about 220,000 miles to about 250,000 miles. This means that the distance is always equal to or more than 220,000 miles, and equal to or less than 250,000 miles. This variation is represented by the variable \(d\).

Formulate the inequality

Formulate the inequality based on the condition that the distance \(d\) is somewhere between 220,000 miles and 250,000 miles. This can be represented as two separate inequalities such as \( d \geq 220,000 \) (The distance is greater than or equal to 220,000 miles) and \( d \leq 250,000 \) (The distance is less than or equal to 250,000 miles). Now combine these together to represent the entire range.

Write the final inequality

Combine Step 2's inequalities into one statement. It is done by getting the range 220,000 miles \(\leq d \leq 250,000\) miles. It represents that \(d\) is always within or equal to the range of 220,000 miles and 250,000 miles.

Key concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and function as the language through which we describe patterns and relationships in mathematics. In essence, an algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. For example, the expression 2x + 3 represents a value that depends on what number x stands for.

When we look at real-world scenarios like the Moon's distance from Earth – which can vary – we can use algebraic expressions to capture this variability. The variable d becomes the symbolic representation of the Moon's distance, allowing us to embody natural fluctuations or ranges in a mathematical model, without needing to pinpoint a single fixed number. This ability to generalize and work with variable quantities is what makes algebra so powerful in modelling and solving real-world problems.

Inequality Representation

Inequality representation is a way of expressing relationships between quantities that are not exactly equal, but instead have a range of possible values. The inequalities \(d \geq 220,000\) and \(d \leq 250,000\) describe such relationships where the actual distance is not known precisely, but is confined within an interval.

These mathematical symbols '\geq' and '\leq' are crucial. They stand for 'greater than or equal to' and 'less than or equal to', respectively. When combined, as in the compound inequality 220,000 \leq d \leq 250,000, they effectively communicate that the value of d, the Moon's distance from Earth, falls within the stated range, including the endpoints. It's essential to always carefully read these symbols, as they dictate the direction of the inequality and consequently, the solution set. Inequalities are a fundamental concept, especially in representing constraints or conditions in various fields such as economics, engineering, and science.

Variable Manipulation

Variable manipulation is a skill that allows us to rework algebraic expressions and inequalities to solve for unknown values or to represent a situation more clearly. It involves operations like adding, subtracting, multiplying, and dividing both sides of an equation or inequality by the same number, and is governed by the rules of algebra.

In the context of writing an inequality for the Moon's distance from Earth, we didn't have to manipulate the variable d because the range was explicitly given. However, if we were to solve an inequality for d or rearrange the inequality to express d in terms of other variables or constants, we would rely on variable manipulation techniques. For example, if we wanted to express the Moon's distance in kilometers instead of miles, we would multiply d by the conversion factor from miles to kilometers. Understanding how to correctly and effectively manipulate variables is critical in finding solutions for algebraic problems and in converting one expression into another to meet specific criteria or simplify a problem.