Problem 15 - Algebra 2

Question

Simplify each expression. $$ \frac{2-\frac{4}{x}}{x-\frac{4}{x}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2}{x+2} \).

Step 1: Identify Numerator and Denominator

The expression to simplify is \( \frac{2-\frac{4}{x}}{x-\frac{4}{x}} \). The numerator is \( 2 - \frac{4}{x} \) and the denominator is \( x - \frac{4}{x} \).

Step 2: Find a Common Denominator

For both the numerator \( 2 - \frac{4}{x} \) and the denominator \( x - \frac{4}{x} \), make the fractions have a common denominator by expressing them as \( \frac{2x - 4}{x} \) and \( \frac{x^2 - 4}{x} \) respectively.

Step 3: Simplify the Fraction

The expression now looks like \( \frac{\frac{2x-4}{x}}{\frac{x^2-4}{x}} \). We can simplify it by multiplying the numerator by the reciprocal of the denominator to obtain: \( \frac{2x-4}{x} \times \frac{x}{x^2-4} = \frac{2x-4}{x^2-4} \).

Step 4: Factor and Simplify Further

Factor the numerator and the denominator: \( 2x-4 = 2(x-2) \) and \( x^2-4 = (x-2)(x+2) \). This gives us: \( \frac{2(x-2)}{(x-2)(x+2)} \).

Step 5: Cancel Common Terms

Cancel the \( (x-2) \) term in both the numerator and the denominator: \( \frac{2}{x+2} \).

Step 6: Concluding Step: Simplified Expression

The expression is simplified to \( \frac{2}{x+2} \).

Key Concepts

Algebraic Fractions

Algebraic fractions are expressions that really aren't too different from regular fractions, except they include variables. If you've worked with fractions before, you'll find algebraic fractions familiar, but with an algebra twist. They're of the form \( \frac{a}{b} \) where either or both \( a \) and \( b \) are algebraic expressions. Just like regular fractions, these expressions follow the same rules of numerator and denominator. Being comfortable manipulating these fractions is key to simplifying them efficiently.

• Think of algebraic fractions as a fraction format where division by zero isn’t allowed.
• Always try to simplify the expression by combining like terms.

Grasping this concept lays the foundation for solving complex algebraic expressions effectively.

Factoring

Factoring is like unlocking a hidden form of expressions and it can turn a tricky algebra problem into a piece of cake. At its heart, factoring involves breaking down an expression into simpler terms, or factors, that when multiplied back together give the original expression. For example, in the step-by-step solution, we see the numerator \( 2x-4 \) is factored to \( 2(x-2) \), and the denominator \( x^2-4 \) is factored to \( (x-2)(x+2) \). This process is crucial since it often reveals terms that can be simplified.

• Look for common factors first as they allow the simplest breakdown.
• Use special factoring formulas like difference of squares, which we used here: \( x^2-4 = (x-2)(x+2) \).

Recognizing these patterns will speed up finding solutions and simplifying the expressions.

Common Denominator

To combine fractions, finding a common denominator is essential. This process simply means adjusting the fractions so they share the same bottom part, making it easier to add, subtract, or simply handle them in some operations. For instance, in our example, we adjusted the expressions to \( \frac{2x-4}{x} \) and \( \frac{x^2-4}{x} \) by using the variable \( x \) as the common denominator.

• When dealing with fractions in numerators and denominators, ensure they both share a common base.
• Adjust expressions carefully without losing any original values or operations.

Mastering this concept helps not only to simplify expressions but also to prepare for higher algebraic operations.

Canceling Common Terms

Once expressions are factored and a common denominator is found, canceling common terms is the next step to simplifying the algebraic fraction. In simpler terms, if the same factor appears in both the numerator and the denominator, you can "cancel" or remove it because it denotes dividing the factor by itself, which equates to one. In our example, the factor \( (x-2) \) appears in both, so by canceling it, we're left with \( \frac{2}{x+2} \).

• Identify matching factors across the numerator and the denominator.
• Ensure that the factors are not equal to zero to avoid undefined expressions.

Canceling common terms reduces complexity greatly and reveals the simplified form of the algebraic fraction.