Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression for \(x=4\). $$\frac{12 x-8}{2 x}$$

Step-by-Step Solution

Answer

The result of the evaluated expression when \(x=4\) is \(5\).

Step 1: Substitution

Replace \(x\) in the expression \( \frac{12 x-8}{2 x} \) with the given value \(4\). This changes the expression to \(\frac{12(4)-8}{2(4)}\), which simplifies to \(\frac{48-8}{8}\).

Step 2: Simplify And Evaluate

Next, proceed to solve the simplified expression \(\frac{48-8}{8}\) by performing the operations within the numerator then the operation between the numerator and the denominator. This gives \(\frac{40}{8}\). Finally, evaluating this gives \(5\).

Key Concepts

Substitution Method

The substitution method in algebra is a common technique used to evaluate expressions, particularly when a variable is given a specific value. This makes solving algebraic equations easier. In this exercise, we started with the expression \( \frac{12x - 8}{2x} \). To use substitution, we replace \(x\) with the given number, \(4\). This transforms our expression into \( \frac{12(4) - 8}{2(4)} \). By carrying out this step-by-step replacement, the expression becomes numerical and ready for simplification. Substitution helps translate the problem from an abstract variable form into a concrete numerical form, which is much easier to work with. Always ensure that each instance of the variable is correctly substituted to avoid errors in subsequent steps.

Simplifying Expressions

Once we've substituted the values in an expression, the next task is to simplify it. Simplification means making the expression as basic as possible so that calculations can be straightforward.Here, after substituting \(x = 4\) into the expression, we got \( \frac{48 - 8}{8} \). The process of simplification involves both performing operations within the numerator and making overall reductions.

• First, solve operations within any grouping symbols, if present.
• Next, perform operations in the correct order—according to the mathematical operation hierarchy: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In our problem, the numerator \(48 - 8\) simplifies directly to \(40\). After simplifying the numerator, you can focus on the operations between the numerator and the denominator.

Numerator and Denominator Operations

Understanding how to handle operations within fractions is key in simplifying expressions. In our example, once you simplify the numerator to \(40\), the next step is focusing on the division part represented by the fraction bar. The fraction \( \frac{40}{8} \) instructs us to divide the numerator by the denominator. Here are the steps:

• Ensure both the numerator and denominator are correctly simplified.
• Perform the division: in this case, dividing \(40\) by \(8\).

By doing this, we obtain the final value of \(5\). Always double-check your division to avoid any simple arithmetic errors. With practice, identifying when and how to simplify will become second nature in solving algebraic expressions. This concludes the evaluation of our expression using operations between the numerator and denominator.