Problem 40 - Precalculus Mathematics for Calculus

Question

Perform the addition or subtraction and simplify. $$\frac{2 x-1}{x+4}-1$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{x-5}{x+4} \).

Step 1: Express Whole Number as Fraction

To perform subtraction with fractions, express the whole number 1 as a fraction with the same denominator as the other term. Here, rewrite 1 as \( \frac{x+4}{x+4} \).

Step 2: Subtract the Fractions

Subtract the fractions: \( \frac{2x-1}{x+4} - \frac{x+4}{x+4} \). Combine them into a single fraction: \( \frac{(2x-1) - (x+4)}{x+4} \).

Step 3: Simplify the Numerator

Simplify the expression in the numerator: \((2x-1) - (x+4) = 2x - 1 - x - 4 = x - 5\). Substitute back into the fraction: \( \frac{x-5}{x+4} \).

Step 4: Confirm and Conclude

Check the result to ensure there are no common factors between \(x-5\) and \(x+4\), concluding that the expression is simplified. The final simplified form is \( \frac{x-5}{x+4} \).

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. Understanding how to work with these is fundamental in algebra. This is because they apply to a multitude of real-life scenarios and higher-level mathematics. In a rational expression, the denominator cannot be zero, as this would make the expression undefined.
For example, in the expression \( \frac{2x-1}{x+4} \), both the numerator \( 2x-1 \) and the denominator \( x+4 \) are polynomials. The variable \( x \) could be any real number except for -4, because that would make the denominator zero.
Recognizing rational expressions and understanding their properties is the first step toward solving problems involving them.

Subtraction of Fractions

Subtracting fractions can seem complex, but it's all about getting the denominators to match. Similar to adding fractions, you need a common denominator to subtract them. In our example, you start by converting the whole number 1 into a fraction with the same denominator as \( \frac{2x-1}{x+4} \). This turns 1 into \( \frac{x+4}{x+4} \), which is equivalent to 1.
This matching of denominators allows you to write the subtraction as one fraction: \( \frac{2x-1}{x+4} - \frac{x+4}{x+4} \). Now you have a common ground to operate the subtraction within the numerator using basic arithmetic skills.

• Ensure each fraction has the same denominator.
• Subtract the numerators while retaining the common denominator.

The key is keeping the denominator consistent, allowing a simple unification of terms in the numerator for subtraction.

Simplification of Expressions

Simplifying expressions means making them as concise as possible. After subtracting in the previous steps, we simplify the numerator \((2x-1) - (x+4)\). Calculating results in \(x - 5\). Substituting this into our final expression gives us \(\frac{x-5}{x+4}\).
The goal in simplification is to reduce the expression to its simplest form by eliminating any like terms or common factors. Here, there are no common factors in the numerator and the denominator. So, \( \frac{x-5}{x+4} \) is simplified completely.
Remember, the more you simplify, the easier it is to understand the underlying patterns in expressions. Checking for anything that can be canceled is always a good final step.