Problem 45 - Algebra 1

Question

SIMPLIFYING RATIONAL EXPRESSIONS Simplify the expression. $$\frac{6 x}{x+1}-\frac{2 x+4}{x+1}$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x - 1}{x+1}\).

Step 1: Understand the problem and set up the expression

We are given the rational expression \(\frac{6x}{x+1} - \frac{2x+4}{x+1}\), since they have the same denominator we can combine them under one fraction.

Step 2: Combine fractions

We can subtract the two fractions by subtracting the numerators, we get \(\frac{6x - (2x + 4)}{x+1}\).

Step 3: Simplify the numerator

Expand the expression in the numerator \(6x - 2x - 4\), you get \(4x - 4\).

Step 4: Final simplification

Finally, divide both terms in the numerator by 4 to simplify the expression, so we get \(\frac{x - 1}{x+1}\).

Key Concepts

Simplifying Expressions

Understanding the process of simplifying expressions is essential when working with rational expressions. Simplifying means making an expression easier to work with by reducing it to its most basic form. In the case of rational expressions, this often involves combining like terms or canceling out shared factors between the numerator and denominator.

Before beginning simplification, identify if all parts of the expression can be further reduced. You want to ensure that the numerator and denominator have been simplified as much as possible. This involves looking out for common factors and knowing how to handle them.

For the expression \(\frac{6 x}{x+1}-\frac{2 x+4}{x+1}\), the common denominator \((x+1)\) allows us to combine terms by focusing on the numerators. This is a common step in simplifying rational expressions, where shared denominators are simplified by merging their numerators.

Combining Fractions

Combining fractions with the same denominator is a straightforward process. It's similar to adding or subtracting numbers, as long as the denominators are the same.

When you combine fractions that share a denominator, simply merge the numerators and keep the denominator unchanged. For the exercise given, the expression \(\frac{6x}{x+1} - \frac{2x+4}{x+1}\) can be combined into a single fraction by subtracting their numerators:

• Keep the common denominator \((x+1)\)
• Subtract the numerators: \(6x - (2x + 4)\)

Expand and simplify the numerator as much as possible. This sets the stage for carrying out further simplifications in the expression, helping isolate the final simplified form.

Combining fractions is crucial because it reduces the number of terms you have to work with, making complex expressions much simpler and easier to evaluate.

Factoring

Factoring is a key tool in algebra, especially when simplifying expressions. Factoring involves breaking down a more complex expression into simpler parts, or factors, that when multiplied together, yield the original expression.

When simplifying rational expressions, you often look for opportunities to factor out common terms from the numerator or the denominator. In the given example, after combining the fractions, the numerator becomes \(4x - 4\). This can be factored further:

• Identify the greatest common factor (GCF) in the numerator, which is 4.
• Factor it out: \(4(x - 1)\).

After factoring, cancel out any common terms from the numerator and denominator. In our example, the common factor 4 allows us to simplify the expression to \(\frac{x-1}{x+1}\), where no further factoring can be done.

Factoring helps identify and cancel out common terms, ultimately reducing the expression to its simplest form, which is much easier to interpret and use in further calculations.