Problem 67 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{4 x}{6-x}+\frac{5}{x-6}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( \frac{4x - 5}{6 - x} \)

Step 1: Change the denominator of the second fraction

Multiply the second fraction by \(-1\) over \(-1\) to remove the negative sign from its denominator. It changes to \(-\frac{5}{6-x}\). Now, both fractions have the same denominator.

Step 2: Combine the fractions

Since the denominators are the same, the fractions can be combined by adding the numerators: \(\frac{4x - 5}{6 - x}\).

Step 3: Simplify the expression

There is no common factor in the numerator and denominator so the expression is already in its simplest form.

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is crucial when working with mathematical equations and simplifying them. In essence, an algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. For example, in the expression \(\frac{4x}{6-x}+\frac{5}{x-6}\), we have variables (x), numbers (4, 5, and 6), and operations (division and addition). These components work together to convey a certain quantity that can change depending on the value of x. It is like a recipe that explains how to combine different ingredients (numbers and variables) to produce a new outcome every time the variable changes.
Simplifying an algebraic expression often requires combining like terms or altering the expression's form to make it easier to understand or solve. This may involve finding common denominators for fraction addition or factoring. The goal is to rewrite the expression in the most straightforward way possible without altering its value. The simpler the form, the easier it becomes to evaluate or use it in further calculations.

Common Denominators

Having a common denominator is vital when it comes to adding or subtracting fractions. A common denominator refers to a shared multiple of the denominators of two or more fractions. It allows you to combine fractions in a meaningful way.
For example, consider the two parts of our original expression, \(\frac{4x}{6-x}\) and \(\frac{5}{x-6}\). These fractions seem different at first glance because their denominators are not identical, yet they are closely related. By recognizing that \(6-x\) and \(x-6\) are opposites, we can manipulate the second fraction such that it shares a denominator with the first, creating a scenario where addition or subtraction becomes possible. Multiplying the numerator and denominator of the second fraction by \( -1 \) gives us a common denominator, which is an essential step for fraction addition (as we'll explore in the next section).
Finding a common denominator typically involves factoring denominators, finding the least common multiple, or occasionally using a simple multiplication trick as seen in our given problem.

Fraction Addition

Fraction addition is one of the fundamental operations in algebra that students need to master. When adding fractions with the same denominator, as we achieved in our exercise by finding a common denominator, the process is straightforward: Simply add the numerators while keeping the denominator the same. This process is akin to adding slices of the same-sized pie together—you're increasing the amount of pie you have without changing the size of each piece.
So taking our common denominator \(6-x\), our next step is to add the numerators from both fractions, leading to the simplified fraction \(\frac{4x-5}{6-x}\). It's important to remember that addition is commutative—that is, the order in which you add numbers does not change the sum. Therefore, when adding the fractions in our exercise, we could've also written the final expression as \(\frac{4x-5}{x-6}\), maintaining its value. However, since the original problem presented the denominator as \(6-x\), it's helpful to keep this form to directly convey that the earlier operation was indeed fraction addition.