Problem 17 - Algebra 2 and Trigonometry

Question

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{1}{2-x}+\frac{2}{x-2} $$

Step-by-Step Solution

Answer

\(\frac{1}{x-2}\), not defined for \(x = 2\).

Step 1: Identify Common Denominator

Since the denominators are \(2-x\) and \(x-2\), notice they are negatives of each other: \(2-x = -(x-2)\). A common denominator can be \(-(x-2)\) or \((x-2)\) by adjusting the signs.

Step 2: Adjust Fractions with Common Denominator

Rewrite the first fraction by changing its sign to match the negative denominator, resulting in: \(-\frac{1}{x-2}\). Thus, the expression becomes:\(-\frac{1}{x-2} + \frac{2}{x-2}\).

Step 3: Combine the Fractions

With a common denominator of \((x-2)\), sum the numerators:\(-1 + 2 = 1\),so the expression simplifies to:\(\frac{1}{x-2}\).

Step 4: Determine Undefined Values

Set the denominator \((x-2)\) to zero to find undefined values:\(x-2 = 0\), solving this gives:\(x = 2\). Therefore, the fraction is undefined at \(x = 2\).

Key Concepts

Addition and Subtraction of Fractions

Adding or subtracting fractions may seem tricky at first, but with some practice, it becomes much easier. The key is to pay attention to the denominators. Individually, each fraction in the expression has its own denominator, and they dictate how we can manipulate those fractions.

When adding or subtracting fractions, always check their denominators first. If they are the same, you can simply add or subtract the numerators. But if they're different, like in our original exercise, we need to adjust them, which we will explain how to do using a common denominator.

So remember, don't let different fractions intimidate you. The strategy is simply to make their mathematical 'bottoms' (denominators) match up, making them ready for you to sum or subtract their tops (numerators) without any fuss.

Common Denominator

Finding a common denominator is crucial when performing operations like addition or subtraction on fractions. The denominator is like a shared base that lets us join two fractions into a single, coherent expression. In our exercise, the denominators were tricky because they were negatives of each other.

• The denominators were \(2-x\) and \(x-2\). Notice these two are related, in fact, they are negatives of one another: \(2-x = -(x-2)\).
• To align them under one umbrella, we chose \(\mathbf{-(x-2)}\) as a common denominator after a small sign adjustment.

This adjustment lets us rewrite and simplify the fraction addition, which moves us one step closer to simplifying the expression completely.

Simplest Form

The simplest form of a fraction means you have reduced it to its most basic version. No further simplification is possible. This often involves canceling out numbers that appear in both the numerator and the denominator or just simplifying the operations already performed.

In our exercise, once we managed to rewrite both fractions to share a common denominator \((x-2)\), we could easily combine them:

• We combined the numerators: \(-1 + 2 = 1\).
• This gave us a simplified fraction: \(\frac{1}{x-2}\).

That fraction is in its simplest form because it has nothing else to subtract or cancel out, leaving it as streamlined as possible for later uses in math.

Undefined Values

Undefined values in fraction expressions occur when the denominator equals zero, as division by zero is undefined in mathematics. It's essential to pinpoint these values because they guide where a function or expression might not work, helping avoid mathematical mishaps.

In the original exercise, after simplifying, our denominator was \(x-2\). So, find out where this becomes zero by setting up an equation:

• \(x-2 = 0\).
• Solving this gives \(x = 2\), meaning the fraction \(\frac{1}{x-2}\) becomes undefined at \(x = 2\).

Identifying these values ensures your calculations stay safe and correct, not leading you into incorrect or undefined results in future calculations.