Problem 28 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$4(x+6)$$

Step-by-Step Solution

Answer

The expression \(4*(x+6)\) can be rewritten without parentheses as \(4*x + 24\).

Step 1: Identify the values in the equation

In our problem, the expression is \(4*(x+6)\). Here, 4 is our \(a\), \(x\) is our \(b\) and 6 is our \(c\). We will apply these values to the distributive property \(a * (b + c) = a * b + a * c\)

Step 2: Implement the distributive property

Now we can apply the distributive property to the given problem. We substitute \(a\), \(b\), and \(c\) from step 1 into the formula. Therefore \(4*(x+6)\) simplifies to \(4*x + 4*6\).

Step 3: Simplify the multiplication

The final step is to perform the multiplication in the expression obtained to get the final answer. This gives us \(4*x + 24\).

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. Variables are symbols that represent unknown values and can stand for numbers that vary. In our original exercise, we have the algebraic expression 4(x+6), which is made up of the number 4 (a coefficient), a variable x, and the number 6 inside the parentheses.

These expressions become the building blocks of algebra and allow us to describe relationships and changes. Understanding algebraic expressions is crucial for solving equations, which is a central theme in intermediate algebra. Algebraic expressions can be manipulated using various properties, like the distributive property, to simplify or solve them.

Simplifying Expressions Using the Distributive Property

The act of simplifying expressions is central to algebra. It involves reducing an expression to its most basic form while maintaining its original value. The distributive property is a valuable tool for this task. It states that for any real numbers a, b, and c, the equation a*(b+c) = a*b + a*c holds true.

When simplifying 4(x+6), we apply this property by distributing the 4 to both x and 6. The result is 4*x + 4*6. Simplifying further, we then multiply to obtain the final simplified form, 4x + 24. By mastering this property, you can tackle a variety of algebraic problems with ease and confidence.

The Role of Intermediate Algebra

The concept of intermediate algebra encompasses a range of topics that build on the fundamentals of basic algebra. It is at this level where students are introduced to more complex expressions and learn to manipulate them using algebraic properties.

Intermediate algebra serves as a bridge to advanced mathematics, requiring a strong understanding of concepts like the distributive property and the ability to simplify expressions. It's all about taking what looks complicated, breaking it down into simpler parts, and finding a solution that makes sense. From factoring polynomials to solving quadratic equations, mastering intermediate algebra paves the way for tackling higher-level math courses.