Problem 50 - Algebra 2

Question

Write two different complex fractions that simplify to \(\frac{x-2}{x+4}\).

Step-by-Step Solution

Answer

Two complex fractions that simplify to \(\frac{x-2}{x+4}\) are \(\frac{x(x-2)}{x(x+4)}\) and \(\frac{2(x-2)}{2(x+4)}\).

Step 1: Generate the first complex fraction

The first complex fraction can be generated simply by multiplying both the numerator and the denominator by the same variable. This will not change the value of the fraction. We can quite easily get the first complex fraction as \(\frac{x(x-2)}{x(x+4)}\).

Step 2: Simplify the first complex fraction

When we simplify \(\frac{x(x-2)}{x(x+4)}\), it will give us \(\frac{x^2-2x}{x^2 +4x}\). We can then divide by the highest power of x to get \(\frac{x-2}{x+4}\).

Step 3: Generate the second complex fraction

Now let's generate the second complex fraction. We can take a constant multiple of our original fraction. Let's multiply both the numerator and the denominator of the fraction by 2. Hence, the second complex fraction is \(\frac{2(x-2)}{2(x+4)}\).

Step 4: Simplify the second complex fraction

When we simplify \(\frac{2(x-2)}{2(x+4)}\), we find that the 2's cancel out, and we are left with our original fraction, \(\frac{x-2}{x+4}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial step when working with mathematical expressions. It involves making a fraction more compact and generally easier to work with.
When you simplify a fraction, you divide both the numerator and the denominator by their greatest common factor. This reduces the fraction to its simplest form, keeping the value of the fraction the same.

• Always look for common factors in the numerator and the denominator.
• This often involves factoring algebraic expressions or canceling constants in terms involving multiplication or division.

For instance, simplifying the complex fraction \( \frac{x(x-2)}{x(x+4)} \) results in \( \frac{x-2}{x+4} \) because we can cancel the \( x \) common to both. This makes calculations and further manipulations easier.

Algebraic Expressions

An algebraic expression consists of variables, numbers, and operations (like addition or multiplication). Unlike simple arithmetic, these expressions use letters to represent numbers, making it possible to solve general problems that work for many different values.
Exploring algebraic expressions involves understanding how to manipulate them. You often need to perform operations such as

• Adding or subtracting expressions
• Factoring expressions to reveal underlying structures
• Simplifying expressions by combining like terms

When handling complex fractions in algebra, like \( \frac{2(x-2)}{2(x+4)} \), it's vital to clearly follow the operations involved in expression manipulation. Simplifying such expressions often makes math problems more approachable and solvable.

Multiplication of Fractions

Multiplying fractions involves multiplying the numerators together and the denominators together. This operation is straightforward but critical when working with algebraic fractions.
For example, to create the complex fraction \( \frac{x(x-2)}{x(x+4)} \), we multiply the original numerator and the denominator by \( x \).

• Ensure both numerators and denominators are correctly factored before multiplying.
• Resulting expressions should be checked for possible simplification.
• Ultimately, the multiplication can often lead to simplification and scale-up effectively.

Understanding multiplication for fractions is important for constructing expressions and solving equations efficiently.

Fraction Equivalence

Fraction equivalence is a concept that ensures different fractions represent the same number or expression. Two fractions are equivalent if they simplify to the same simplest form.
This property is pivotal when transforming fractions into complex forms to achieve a specific representation. For instance, \( \frac{2(x-2)}{2(x+4)} \) appears different but is equivalent to \( \frac{x-2}{x+4} \) because their simplified forms are identical.

• Start by understanding that equivalent fractions can be generated by multiplying or dividing both the numerator and the denominator by the same non-zero number.
• Use equivalence to switch between different forms as needed, especially in algebraic manipulations.
• Verification of equivalence often involves simplification to check if they reduce to the same expression.

Harnessing the power of fraction equivalence allows deeper structural insights into algebraic problems and solutions.