Problem 119 - Introductory Algebra for College Students

Question

Find the missing rational expression. $$\frac{4}{x-2}-\frac{2 x+8}{(x-2)(x+1)}$$

Step-by-Step Solution

Answer

The missing rational expression is \( \frac{x-2}{(x-2)(x+1)} \).

Step 1: Make the denominators the same

In order to operate between fractions, their denominators must be the same. Currently, the second fraction has \((x-2)(x+1)\) in the denominator, so the first fraction should be rewritten as \( \frac{4(x+1)}{(x-2)(x+1)} \) to have the same denominator.

Step 2: Subtract the fractions

Now that both fractions have the same denominator of \((x-2)(x+1)\), we can subtract them: \( \frac{4(x+1) - (2x+8)}{(x-2)(x+1)} .\)

Step 3: Simplify the subtraction result

Simplify the numerator: \( \frac{4(x+1) - (2x+8)}{(x-2)(x+1)} = \frac{4x+4-2x-8}{(x-2)(x+1)} .\) This simplifies further to \( \frac{2x-4}{(x-2)(x+1)} .\)

Step 4: Simplify result further

Notice that the numerator \(2x - 4\) is divisible by 2. To simplify further, divide both terms in the numerator by 2 to get \( \frac{x-2}{(x-2)(x+1)} .\)

Key Concepts

Subtracting Fractions

Rational expressions often involve fractions, which require careful handling when subtracting. To subtract fractions, their denominators need to be identical. Think of subtracting fractions like combining slices of pizza. You can only subtract if they're cut into the same-sized pieces. In the given exercise, the first fraction \( \frac{4}{x-2} \) and the second fraction \( \frac{2x+8}{(x-2)(x+1)} \) have different denominators.

• To handle this, we turn both fractions into ones with matching denominators.
• This way, the subtraction becomes manageable and straightforward.

Once they are under a shared denominator, you simply subtract the numerators and keep the common denominator. Remember, the key to subtracting fractions is matching the denominators first, then moving on to the numerators.

Simplifying Algebraic Expressions

Once you've subtracted the fractions, the next step is to simplify the resulting algebraic expression. Simplification makes things easier and clearer to handle.

• The expression \( \frac{4(x+1) - (2x+8)}{(x-2)(x+1)} \) first needs the numerators subtracted.
• This means distributing and combining like terms.

Break it down step-by-step: - Distribute 4 in the first part: \( 4(x+1) \) becomes \( 4x + 4 \).- Subtract the second part: \( 2x + 8 \).- Combine the like terms: \( 4x + 4 - 2x - 8 \) to get \( 2x - 4 \).Simplifying expressions leads to a cleaner, more elegant formula that is often easier to interpret and further manipulate.

Common Denominator

Finding a common denominator is essential when working with multiple fractions. It allows for straightforward arithmetic operations like addition or subtraction. In this context, the common denominator means finding a shared base that all fractions can be expressed in terms of.

• For the equation \( \frac{4}{x-2} - \frac{2x+8}{(x-2)(x+1)} \), the second fraction already has the denominator \((x-2)(x+1)\).
• Multiply the first fraction's numerator by \(x+1\) to match this format.

By rewriting \( \frac{4}{x-2} \) as \( \frac{4(x+1)}{(x-2)(x+1)} \), both fractions now have the common denominator \((x-2)(x+1)\). This uniformity is crucial because it allows you to focus solely on the numerators for subtraction.