Problem 46 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Whole Number as a Fraction

We have the expression \( \frac{2x-1}{x+4} - 1 \). The whole number \(1\) can be rewritten as a fraction with a denominator of \(x+4\), which gives us \(\frac{x+4}{x+4}\). This way, both terms have a common denominator.

Step 2: Perform the Subtraction

Subtract the fraction \(\frac{x+4}{x+4}\) from the fraction \(\frac{2x-1}{x+4}\). Write the subtraction as: \[\frac{2x-1}{x+4} - \frac{x+4}{x+4}\]. Combine the numerators over the common denominator: \[\frac{(2x-1)-(x+4)}{x+4}\].

Step 3: Simplify the Numerator

Simplify the expression \((2x-1)-(x+4)\):1. Distribute the negative sign: \(2x-1-x-4\).2. Combine like terms: \((2x-x) + (-1-4) = x - 5\).This gives us the simplified expression \(\frac{x-5}{x+4}\).

Step 4: Conclusion

There are no common factors in the numerator and denominator, so the expression is already in its simplest form. Thus, the final simplified expression is \(\frac{x-5}{x+4}\).

Key Concepts

Fraction Subtraction

Subtraction of fractions may seem tricky at first, but don't worry! The key is ensuring the denominators are the same. This makes it simpler to only focus on the numerators for subtraction.

In the problem, we have a fraction, \( \frac{2x-1}{x+4} \), and we want to subtract 1. Because 1 is not originally a fraction, we need to convert it to a fraction form with the same denominator, \( x+4 \), so it becomes \( \frac{x+4}{x+4} \).

Once both parts of the expression have the same denominator, we can keep the denominator \( x+4 \) unchanged while subtracting the numerators. In our case:

• First numerator: \(2x - 1\)
• Second numerator: \(x + 4\)

Now, perform the subtraction \((2x - 1) - (x + 4)\). This leads us directly to the next step: Simplification.

Simplifying Expressions

After setting up the numerator for subtraction, it's time to simplify. The objective here is to combine like terms, ensuring the expression is as neat and condensed as possible.

Begin by distributing the negative sign across terms in the second numerator:

• \(2x - 1 - x - 4\)

This step might sound simple, but it's crucial. Negative signs can change everything!

Once the distribution is done, combine like terms:

• Combine \(2x\) and \(-x\) to get \(x\).
• Combine \(-1\) and \(-4\) to get \(-5\).

Thus, the simplified form of the expression is \(x - 5\). This process helps in reducing clutter and makes it easier to understand the essence of the expression. Simplifying ensures the final answer is clean and fully reduced.
This leads us to the importance of understanding the common denominator in subtraction.

Common Denominator

Achieving a common denominator is the foundation of easily handling problems with fractions, especially subtraction. Without a common denominator, subtracting fractions turns into a more arduous task.

The common denominator allows you to align fractions, focusing solely on the numerators while keeping the denominator static.

• For our problem, the denominator \(x+4\) was shared between both fractions.
• Rewriting 1 as a fraction \(\frac{x+4}{x+4}\) ensures both parts of the expression play nice together.

By understanding and applying the common denominator, you ensure every step flows seamlessly into the next.

This understanding allows anyone dealing with algebraic fractions to subtract, add, and even compare them without complication. It's not just about solving the problem but mastering this essential concept that empowers you to tackle any fraction-related challenge with confidence.