Problem 8 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=4\). $$\frac{36}{x}$$

Step-by-Step Solution

Answer

The evaluated expression for \(x=4\) is 9.

Step 1: Substitution

Replace \(x\) in the expression \(\frac{36}{x}\) with 4.

Step 2: Compute the Expression

Calculate \(\frac{36}{4}\) which equals to 9

Key Concepts

Substitution Method

The substitution method is a fundamental technique in evaluating algebraic expressions. It involves replacing variables with given numerical values to simplify and solve the expression. Essentially, it transforms an algebraic equation into a more manageable arithmetic problem.

For instance, given a problem where you need to evaluate \(\frac{36}{x}\) for \(x=4\), the method dictates that you replace every \(x\) in the expression with the number 4. This step directly leads to an arithmetic calculation, turning a variable-laden equation into a straightforward numerical fraction to be simplified.

Arithmetic Operations

Arithmetic operations are the bread and butter of mathematics, including addition, subtraction, multiplication, and division. When solving algebraic expressions, especially after applying the substitution method, these operations come into play to simplify and reach the final answer.

Using the expression \(\frac{36}{x}\) with \(x=4\), once substitution is done, the evaluation comes down to division, one of the core arithmetic operations. The simplicity of arithmetic allows the initial algebra problem to be decoded into a problem of dividing 36 by 4 to arrive at the answer.

Algebra Problem Solving

Algebra problem solving is essentially about finding unknown values or simplifying expressions using various techniques. It typically involves a clear understanding of variable manipulation, the order of operations, and, quite often, the use of the substitution method.

In solving \(\frac{36}{x}\) for \(x=4\), problem-solving skills are crucial. After substituting 4 into the equation, solving the problem successfully requires the knowledge of how to divide numbers correctly. It is a process that demonstrates how algebra can be distilled into basic arithmetic.

Intermediate Algebra

Intermediate Algebra spans a wide range of mathematical concepts, building on elementary algebra and preparing students for more advanced studies in mathematics. It includes various functions, polynomial and rational expressions, and systems of equations.

In the context of evaluating \(\frac{36}{x}\) when \(x=4\), this falls under the early stages of intermediate algebra. This area of study emphasizes the application of fundamental algebraic principles to solve equations and enhance the learner's ability to tackle more complex problems in the future.