One Less Than Twice A Number

One Less Than Twice a Number: Unraveling the Mathematical Mystery

The seemingly simple phrase, "one less than twice a number," hides a surprisingly rich world of mathematical concepts and applications. This seemingly basic algebraic expression opens doors to understanding fundamental principles, solving equations, and even delving into more advanced mathematical territories. Let's unravel the mystery behind this phrase and explore its various facets.

Understanding the Core Concept

At its heart, "one less than twice a number" is a concise way of expressing a mathematical relationship. Let's break it down step-by-step:

• A number: This represents an unknown value, often symbolized by a variable like x, y, or n.
• Twice a number: This means multiplying the number by 2 (2 * x, 2y, or 2n).
• One less than twice a number: This signifies subtracting 1 from the result of doubling the number (2x - 1, 2y - 1, or 2n - 1).

Therefore, the phrase translates directly into the algebraic expression: 2x - 1 (using 'x' as our variable). This simple expression forms the foundation for numerous mathematical problems and applications.

Solving Equations Involving "One Less Than Twice a Number"

The real power of this expression comes into play when it's part of an equation. Let's explore some examples:

Example 1: Finding the Number

Problem: One less than twice a number is 7. Find the number.

Solution:

1. Translate the problem into an equation: We can represent this problem as: 2x - 1 = 7
2. Solve for x:
   • Add 1 to both sides: 2x = 8
   • Divide both sides by 2: x = 4

Therefore, the number is 4.

Example 2: Word Problem Application

Problem: John's age is one less than twice his sister's age. If John is 15 years old, how old is his sister?

Solution:

1. Define variables: Let's use 's' to represent the sister's age.
2. Translate the problem into an equation: John's age (15) is one less than twice his sister's age: 15 = 2s - 1
3. Solve for s:
   • Add 1 to both sides: 16 = 2s
   • Divide both sides by 2: s = 8

Therefore, John's sister is 8 years old.

Example 3: Incorporating Inequalities

The expression can also be used within inequalities.

Problem: One less than twice a number is greater than 5. Find the range of possible values for the number.

Solution:

1. Translate the problem into an inequality: 2x - 1 > 5
2. Solve for x:
   • Add 1 to both sides: 2x > 6
   • Divide both sides by 2: x > 3

Therefore, the number must be greater than 3.

Advanced Applications and Extensions

While the core concept is relatively straightforward, the expression "one less than twice a number" can be incorporated into more complex mathematical scenarios:

Quadratic Equations

Imagine a scenario where the expression is part of a quadratic equation:

Problem: The area of a rectangle is given by the expression x(2x - 1) = 30, where x represents the width. Find the width of the rectangle.

Solution:

1. Expand the equation: 2x² - x = 30
2. Rearrange into standard quadratic form: 2x² - x - 30 = 0
3. Solve the quadratic equation: This can be done using factoring, the quadratic formula, or other methods. The solutions for x might be positive and negative; however, only positive solutions have a real-world meaning (in this case, width).

This demonstrates how the seemingly simple expression can lead to more complex problem-solving involving quadratic equations.

Sequences and Series

The expression could also define a term in a sequence or series. For example, a sequence could be defined as: a_n = 2n - 1. This represents an arithmetic sequence where each term is one less than twice its position in the sequence.

Functions and Graphing

The expression can be represented as a function: f(x) = 2x - 1. This linear function can then be graphed, allowing for visual representation of the relationship between the input (x) and the output (f(x)). Analyzing the graph can reveal important properties of the function, such as its slope and y-intercept.

Real-World Applications

While seemingly abstract, the concept of "one less than twice a number" has practical applications in various fields:

• Physics: Many physical phenomena can be modeled using linear equations, where this expression could represent a component of a larger formula.
• Engineering: Similar to physics, engineering often involves calculations where linear relationships are crucial, and this expression could form a part of these calculations.
• Economics: Economic models often use linear equations to represent relationships between variables. This expression could appear within such models.
• Computer Science: Programming frequently involves the manipulation of numerical data, and understanding this expression helps in creating algorithms and code for calculations.

Expanding Mathematical Understanding

The simple phrase, "one less than twice a number," serves as a gateway to a deeper understanding of fundamental mathematical concepts. It highlights the importance of:

• Variable representation: Using variables to represent unknown quantities is crucial for solving mathematical problems.
• Equation solving: Mastering the techniques for solving equations is essential for tackling more advanced mathematical concepts.
• Algebraic manipulation: The ability to manipulate algebraic expressions is a fundamental skill in mathematics.
• Problem-solving: The application of mathematical concepts to real-world problems enhances problem-solving skills.

By exploring this seemingly simple expression, we gain insights into the broader world of algebra and its practical applications. It underscores the beauty and power of mathematics in explaining and solving problems in various aspects of our lives. The ability to translate verbal descriptions into mathematical expressions is a key skill, and "one less than twice a number" provides an excellent example of this crucial translation. The practice of solving problems involving this expression builds a strong foundation for tackling more complex mathematical challenges in the future. From simple equations to more advanced applications, the exploration of this expression reveals the interconnectedness and elegance of mathematical concepts. Remember, the journey of mathematical understanding is a continuous process of exploration and discovery, and this simple phrase serves as a perfect starting point.