3 Less Than The Product Of 8 And A Number

3 Less Than the Product of 8 and a Number: A Deep Dive into Mathematical Expressions

This seemingly simple phrase, "3 less than the product of 8 and a number," hides a wealth of mathematical concepts. Understanding how to translate this phrase into an algebraic expression, solve related equations, and apply it to real-world scenarios is crucial for developing strong mathematical reasoning skills. This article will explore this phrase in detail, covering various aspects from basic algebra to more advanced problem-solving techniques.

Understanding the Components

Before diving into the algebraic representation, let's break down the phrase into its individual components:

• A number: This represents an unknown value. In algebra, we typically represent unknown values with variables, most commonly x. However, other letters like y, n, or even more descriptive variables can be used depending on the context of the problem.

• The product of 8 and a number: "Product" signifies multiplication. Therefore, the product of 8 and a number (represented by x) is written as 8x or 8x.

• 3 less than: This indicates subtraction. "3 less than" a quantity means subtracting 3 from that quantity.

Translating the Phrase into an Algebraic Expression

Putting it all together, "3 less than the product of 8 and a number" translates to the algebraic expression: 8x - 3. This expression represents a mathematical relationship where the value of the entire expression depends on the value of x.

Exploring Different Values of x

Let's explore how the value of the expression changes with different values of x:

• If x = 1: 8(1) - 3 = 5
• If x = 2: 8(2) - 3 = 13
• If x = 0: 8(0) - 3 = -3
• If x = -1: 8(-1) - 3 = -11
• If x = 5: 8(5) - 3 = 37

This demonstrates that the expression 8x - 3 produces a wide range of outputs depending on the input value of x. This is a fundamental concept in algebra: variables allow for the representation of dynamic relationships.

Solving Equations Involving the Expression

Often, we encounter situations where the expression 8x - 3 is equated to a specific value. This forms an algebraic equation, which we can solve to find the value of x. Let's consider a few examples:

Example 1: 8x - 3 = 19

1. Add 3 to both sides: 8x = 22
2. Divide both sides by 8: x = 22/8 = 11/4 = 2.75

Therefore, if the expression 8x - 3 equals 19, the value of x is 2.75.

Example 2: 8x - 3 = -11

1. Add 3 to both sides: 8x = -8
2. Divide both sides by 8: x = -1

In this case, if the expression equals -11, the value of x is -1.

Example 3: Solving for x when 8x - 3 is greater than 20

This introduces inequalities. We need to solve for x when 8x - 3 > 20:

1. Add 3 to both sides: 8x > 23
2. Divide both sides by 8: x > 23/8 = 2.875

This means that for the expression 8x - 3 to be greater than 20, x must be greater than 2.875.

Real-World Applications

While the expression itself might seem abstract, it has many practical applications:

• Calculating earnings: Imagine earning $8 per hour (8x) and having $3 deducted for taxes (-3). The expression 8x - 3 could represent your net earnings after taxes.

• Calculating profits: A business might have a profit of $8 per unit sold (8x) but faces fixed costs of $3 (-3). The expression represents the net profit after deducting the fixed costs.

• Modeling physical quantities: In physics, an object's velocity might be modeled by an equation involving this expression, where x represents time.

• Financial modeling: The expression could be part of a more complex formula in finance calculations, involving interest, principal, or other financial variables.

Advanced Concepts and Extensions

This simple expression can form the foundation for more complex mathematical explorations. For example:

• Quadratic Equations: If we square the expression, we get (8x - 3)², leading us to quadratic equations, which require different solving techniques.

• Functions: The expression 8x - 3 can be represented as a function, f(x) = 8x - 3. This allows for analysis of the function's properties, such as its slope and y-intercept.

• Systems of Equations: We could have a system of equations where 8x - 3 is one equation, along with another equation involving x (and perhaps other variables). Solving these systems requires advanced techniques like substitution or elimination.

• Calculus: Derivatives and integrals of the expression can be calculated, revealing insights into its rate of change and accumulated area under its curve.

Word Problems and Practical Application

Let's look at some word problems that use this expression:

Problem 1: Sarah earns $8 per hour babysitting. She has to pay a $3 fee for using the online babysitting platform. Write an expression for her total earnings, and calculate her earnings if she worked for 5 hours.

• Expression: 8x - 3, where x is the number of hours worked.
• Solution: 8(5) - 3 = $37

Problem 2: A store sells candles for $8 each. The store's fixed costs are $3. Write an expression for the store's profit, and determine the profit if they sell 10 candles.

• Expression: 8x - 3, where x is the number of candles sold.
• Solution: 8(10) - 3 = $77

Problem 3: A rectangular garden has a length that is 3 less than 8 times its width (x). Write an expression for the garden's length. If the width is 2 meters, what is the length?

• Expression: 8x - 3, where x is the width.
• Solution: 8(2) - 3 = 13 meters.

Conclusion: A Foundation for Further Learning

The seemingly simple phrase, "3 less than the product of 8 and a number," serves as a powerful entry point into the world of algebra. By understanding its translation into an algebraic expression, solving related equations, and applying it to real-world scenarios, we gain a deeper appreciation for the power and versatility of mathematics. This fundamental concept lays the groundwork for more advanced topics and problem-solving strategies, emphasizing the importance of careful translation and the interpretation of mathematical language. Mastering this seemingly simple expression builds a strong foundation for future mathematical success.