Problem 49 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x-4}-2}{\frac{4}{x+5}-4} $$

Step-by-Step Solution

Answer

\( \frac{(9-2x)(x+5)}{(x-4)[-4(x+4)]} \)

Step 1 - Simplify the numerator

Start by simplifying the numerator \( \frac{1}{x-4} - 2 \). First, express 2 as \( \frac{2(x-4)}{x-4} \) to have a common denominator. Thus, the numerator becomes \(\frac{1 - 2(x-4)}{x-4} \).

Step 2 - Combine the terms in the numerator

Simplify the expression in the numerator: \(\frac{1 - 2(x-4)}{x-4} = \frac{1 - 2x + 8}{x-4} = \frac{9-2x}{x-4} \).

Step 3 - Simplify the denominator

Simplify the denominator \(\frac{4}{x+5} - 4\). Express 4 as \( \frac{4(x+5)}{x+5} \) to have a common denominator. Then, the denominator becomes \(\frac{4 - 4(x+5)}{x+5} \).

Step 4 - Combine the terms in the denominator

Simplify the expression in the denominator: \(\frac{4 - 4(x+5)}{x+5} = \frac{4 - 4x - 20}{x+5} = \frac{-4x - 16}{x+5} = \frac{-4(x+4)}{x+5} \).

Step 5 - Combine the simplified numerator and denominator

Now combine the simplified numerator and denominator: \(\frac{\frac{9-2x}{x-4}}{\frac{-4(x+4)}{x+5}} = \frac{9-2x}{x-4} \times \frac{x+5}{-4(x+4)} \).

Step 6 - Simplify the final expression

Simplify by multiplying the fractions: \(\frac{(9-2x)(x+5)}{(x-4)[-4(x+4)]} \). This is the simplified form of the original expression.

Key Concepts

Simplifying Fractions

Simplifying fractions is about expressing a fraction in its simplest form. You do this by finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by their GCD.
In the given exercise, we simplify both the numerator and the denominator separately. This helps to break down complex fractions into simpler parts and makes it easier to solve them.
After breaking the problem down, you do the same kind of operations with each part of the fraction.

Common Denominators

When combining fractions, it's crucial to have a common denominator. This means that the denominators (the bottom parts of fractions) must be the same.
For example, in the given problem, the numerator \(\frac{1}{x-4} - 2\) was combined by converting '2' into \(\frac{2(x-4)}{x-4}\).
This gives both fractions the same bottom part, making it easier to subtract them.

• First, make sure both fractions have the same denominator.
• Convert whole numbers to fractions with a common denominator.

This tactic applies to both the numerator and denominator of the larger fraction.

Numerator and Denominator Operations

When simplifying fractions, you often need to perform operations separately on the numerator and denominator.
For instance, in the exercise, we simplified the numerator \(\frac{1}{x-4} - 2\) and the denominator \(\frac{4}{x+5} - 4\) step by step. Here are the steps:

• Rewrite each fraction to have a common denominator.
• Perform subtraction or addition on the numerators.
• Simplify the resulting fraction by combining like terms.

Finally, you put the simplified numerator over the simplified denominator and simplify further if possible.

Algebraic Expressions

An algebraic fraction includes variables (like x) and constants. Simplifying these can be tricky, but following systematic steps makes it easier.
In this exercise, the algebraic expression of the numerator \(\frac{1 - 2(x-4)}{x-4}\) and the denominator \(\frac{4 - 4(x+5)}{x+5}\) were simplified.

• First, distribute and combine like terms.
• Factorize common terms if necessary.
• Finally, simplify the remaining expression.

Using these techniques, you can simplify even complex algebraic fractions.