Five Less Than Four Times A Number

Five Less Than Four Times a Number: A Deep Dive into Algebraic Expressions

This seemingly simple phrase, "five less than four times a number," hides a wealth of mathematical concepts. It's a gateway to understanding algebraic expressions, equation solving, and even more advanced topics like inequalities and functions. This comprehensive guide will explore this phrase in detail, revealing its underlying structure, practical applications, and the broader mathematical landscape it represents.

Understanding the Phrase: Deconstructing the Expression

The phrase "five less than four times a number" is a verbal description of an algebraic expression. Let's break it down piece by piece:

• A number: This represents an unknown value, typically denoted by a variable, most commonly 'x'. However, any letter can be used.

• Four times a number: This translates directly to 4 * x, or more concisely, 4x. This indicates multiplication.

• Five less than: This means we subtract 5 from the previous result.

Therefore, the complete algebraic expression for "five less than four times a number" is 4x - 5. It's crucial to understand the order of operations here; subtraction comes after multiplication. Writing it as 5 - 4x would represent a completely different expression ("five less four times a number").

The Importance of Order of Operations (PEMDAS/BODMAS)

Remember the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This dictates the sequence in which operations must be performed. In our expression, multiplication (4x) is performed before subtraction (-5). Ignoring this order leads to incorrect results.

Working with the Expression: Examples and Applications

Now that we understand the expression, let's explore some practical applications and examples:

Example 1: Finding the Value of the Expression

Let's say 'x' equals 3. Substituting this into our expression:

4x - 5 = 4(3) - 5 = 12 - 5 = 7

Therefore, when x = 3, the expression "five less than four times a number" equals 7.

Example 2: Solving an Equation

Let's create an equation using our expression:

4x - 5 = 11

To solve for x, we need to isolate 'x' on one side of the equation. We can do this by following these steps:

1. Add 5 to both sides: 4x - 5 + 5 = 11 + 5 => 4x = 16
2. Divide both sides by 4: 4x / 4 = 16 / 4 => x = 4

Therefore, the solution to the equation 4x - 5 = 11 is x = 4.

Example 3: Real-World Application: Geometry

Imagine you're calculating the perimeter of a rectangle. Let's say the length is four times the width (x), and the perimeter is 22 units less than 4 times the length. You can represent the width as 'x' and the length as '4x'. If the perimeter is 50 units, you can set up an equation:

2(4x) + 2(x) = 50

Simplifying:

8x + 2x = 50 10x = 50 x = 5

Thus, the width is 5 units, and the length is 20 units. This demonstrates how our initial expression can be a component of more complex real-world problems.

Example 4: Word Problems and Translation

Many word problems require translating verbal descriptions into algebraic expressions. Consider this: "John earns four times his brother's weekly salary, and he spends five dollars less than his earnings each week." Let’s denote his brother’s weekly salary as 'y'. Then John's weekly earnings are '4y', and his weekly spending is '4y - 5'. This shows how easily our phrase translates into real-life scenarios.

Expanding the Concept: Inequalities and Functions

Our simple expression can be incorporated into more advanced mathematical concepts:

Inequalities

Instead of an equation, we can create an inequality:

4x - 5 > 10

This reads as "five less than four times a number is greater than 10." Solving this inequality involves the same steps as solving an equation, but with one crucial difference: when multiplying or dividing by a negative number, you must reverse the inequality sign. Solving this example:

1. Add 5 to both sides: 4x > 15
2. Divide both sides by 4: x > 15/4 or x > 3.75

This means any value of x greater than 3.75 satisfies the inequality.

Functions

Our expression can also define a function. A function is a relationship between inputs (x) and outputs (y). We can write our expression as a function:

f(x) = 4x - 5

This means the function 'f' takes an input 'x', performs the operations (4x - 5), and produces an output, f(x). This allows for exploring the behavior of the expression across a range of x values, paving the way for concepts like graphing and function analysis.

Visual Representation: Graphing the Function

Graphing the function f(x) = 4x - 5 provides a visual representation of its behavior. The graph will be a straight line with a slope of 4 (meaning it rises 4 units for every 1 unit increase in x) and a y-intercept of -5 (meaning it crosses the y-axis at -5). This visual representation offers a powerful way to understand the relationship between the input and output values.

Advanced Applications: Systems of Equations and Beyond

The expression can form part of more complex mathematical structures:

• Systems of Equations: Imagine a scenario involving two related expressions. You might have a system of equations where our expression is one of the equations, alongside another. Solving such a system would involve finding the values of x that satisfy both equations simultaneously. This might involve techniques like substitution or elimination.

• Polynomial Expressions: Our expression can be integrated into more intricate polynomial expressions, which involve higher powers of x. For example, it could be a term within a quadratic equation or a higher-degree polynomial.

• Calculus: In calculus, this simple linear function can be used to illustrate fundamental concepts such as derivatives (the rate of change of the function) and integrals (the area under the curve).

Conclusion: The Significance of Simple Expressions

The seemingly trivial phrase, "five less than four times a number," serves as a fundamental building block in algebra and beyond. Understanding its structure, manipulating it algebraically, and applying it to various mathematical contexts demonstrates the power of translating simple word problems into precise mathematical expressions and then utilizing a range of techniques to solve equations, inequalities, and analyse functions. It provides a firm foundation for tackling more complex mathematical challenges, showcasing the significant role that even basic algebraic expressions play in a wider mathematical landscape. Mastering this concept opens doors to a richer understanding of mathematical principles and their applications in countless fields.