3 Is Subtracted From Three Times A Number.

3 is Subtracted from Three Times a Number: A Deep Dive into Mathematical Expressions and Problem Solving

This seemingly simple phrase, "3 is subtracted from three times a number," hides a wealth of mathematical concepts and problem-solving techniques. This article will explore this phrase in detail, examining its translation into algebraic expressions, solving various problems based on it, and delving into the broader mathematical principles involved. We'll also touch upon real-world applications and explore how understanding this concept can improve your overall mathematical abilities.

Understanding the Phrase: From Words to Algebra

The core of this problem lies in translating the English phrase into a mathematical expression. Let's break it down step by step:

• "A number": This represents an unknown value, typically denoted by a variable, usually 'x' or another letter.

• "Three times a number": This translates directly to 3 * x, or more simply, 3x. This indicates multiplication.

• "3 is subtracted from three times a number": This means we take 3 and subtract it from 3x. The order is crucial here. It's not 3x - 3, but rather 3x - 3.

Therefore, the complete algebraic expression representing the phrase is 3x - 3. This seemingly simple expression opens the door to a wide range of mathematical possibilities.

Solving Equations: Different Scenarios

Now that we have our algebraic expression, let's explore how it's used in various equations. The most common scenario involves setting this expression equal to another value, creating an equation that we can solve for 'x'.

Scenario 1: Finding the Number

Let's say the problem states: "3 is subtracted from three times a number, and the result is 12. Find the number."

This translates to the equation: 3x - 3 = 12

To solve for 'x', we follow these steps:

1. Add 3 to both sides: This isolates the term with 'x'. The equation becomes 3x = 15.

2. Divide both sides by 3: This solves for 'x'. The solution is x = 5.

Therefore, the number is 5. We can check our work: 3 * 5 - 3 = 12. The equation holds true.

Scenario 2: Inequalities

The expression can also be used in inequalities. For example: "3 is subtracted from three times a number, and the result is greater than or equal to 6."

This translates to the inequality: 3x - 3 ≥ 6

Solving this inequality:

1. Add 3 to both sides: 3x ≥ 9

2. Divide both sides by 3: x ≥ 3

This means the number ('x') is greater than or equal to 3. This isn't a single solution but a range of solutions.

Scenario 3: Word Problems and Real-World Applications

Let's consider a real-world example. Imagine a scenario where a phone company charges $3 for a base service fee, plus $3 per minute of usage. The total cost (C) can be represented as: C = 3x + 3, where x is the number of minutes.

This is a slightly different equation, but the core concept of a base value and a variable cost per unit remains the same. If the total cost is $15, you can easily solve for 'x' (the number of minutes used).

Another example could be related to profit calculations in business: If a company makes a profit of $3 for each product sold, and has a fixed cost of $3, its net profit could be modeled using a similar equation. Understanding these mathematical relationships is crucial in various fields.

Expanding the Concepts: Further Exploration

Beyond simple equations and inequalities, the expression "3x - 3" can be used in more complex mathematical contexts:

Functions and Graphing

The expression can define a function: f(x) = 3x - 3. This function can be graphed on a coordinate plane, revealing its linear nature (a straight line with a slope of 3 and a y-intercept of -3). Understanding the graph allows for visual interpretation of the function's behavior.

Polynomials and Higher-Order Equations

While the expression itself is linear, it can be incorporated into more complex polynomial equations. For example, it could be a term within a quadratic equation, leading to more involved solution methods (e.g., using the quadratic formula).

Calculus and Rates of Change

In calculus, the expression could represent a rate of change. The derivative of 3x - 3 is simply 3, indicating a constant rate of change. This concept is fundamental in understanding how quantities change over time.

Importance of Mastering Algebraic Expressions

Mastering the ability to translate phrases like "3 is subtracted from three times a number" into algebraic expressions is fundamental to success in mathematics and its applications. It lays the foundation for:

• Problem-solving: The ability to model real-world situations using mathematical equations is a powerful tool for solving problems in various fields.

• Critical thinking: Breaking down complex word problems into smaller, manageable parts requires careful analysis and critical thinking skills.

• Abstract reasoning: Algebra involves working with abstract concepts and symbols, fostering the development of abstract reasoning abilities.

Conclusion: A Building Block of Mathematics

The seemingly simple phrase, "3 is subtracted from three times a number," serves as a foundational building block in mathematics. Understanding its translation into algebraic expressions, solving equations and inequalities based on it, and exploring its applications in various mathematical contexts are crucial for developing a strong mathematical foundation. This knowledge empowers you to solve a wide range of problems, from basic arithmetic to complex real-world scenarios, highlighting the importance of mastering fundamental algebraic concepts. The skills honed through working with such expressions extend far beyond the classroom, proving invaluable in numerous fields and contributing to overall critical thinking abilities. Remember that practice is key, so continue tackling various problems using this foundational concept, gradually increasing their complexity to solidify your understanding.