Problem 38 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{2 x}{x+2}-\frac{x+1}{x-3} $$

Step-by-Step Solution

Answer

\[\frac{x^2 - 9x - 2}{(x+2)(x-3)}\]

Step 1: Find a Common Denominator

To subtract fractions, we must first find a common denominator. The denominators here are \(x+2\) and \(x-3\). The common denominator will be the product of these two expressions: \[(x+2)(x-3)\].

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite the first fraction by multiplying both numerator and denominator by \((x-3)\): \[\frac{2x(x-3)}{(x+2)(x-3)} = \frac{2x(x-3)}{(x+2)(x-3)}\]Rewrite the second fraction by multiplying both numerator and denominator by \((x+2)\): \[\frac{(x+1)(x+2)}{(x-3)(x+2)} = \frac{(x+1)(x+2)}{(x-3)(x+2)}\].

Step 3: Subtract the Fractions

Now subtract the two fractions: \[\frac{2x(x-3) - (x+1)(x+2)}{(x+2)(x-3)}\]First, let's expand the numerators: \[2x(x-3) = 2x^2 - 6x\]and \[(x+1)(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2\].Substitute back into the equation:\[\frac{2x^2 - 6x - (x^2 + 3x + 2)}{(x+2)(x-3)}\].

Step 4: Simplify the Numerator

Distribute the negative sign across the second term of the numerator:\[2x^2 - 6x - x^2 - 3x - 2\].Now combine like terms:\[2x^2 - x^2 = x^2\]\[-6x - 3x = -9x\]Leaving:\[x^2 - 9x - 2\].

Step 5: Final Simplified Expression

The simplified form of the expression inside the numerator is \[x^2 - 9x - 2\].So, the final expression for the subtraction is:\[\frac{x^2 - 9x - 2}{(x+2)(x-3)}\].This expression is in its simplest form.

Key Concepts

Common Denominator

When you work with fractions in algebra, finding a common denominator is essential for adding or subtracting them. A common denominator is a shared multiple of the original denominators, allowing you to combine the fractions effectively. In this exercise, the denominators are \(x+2\) and \(x-3\).

We achieve a common ground by multiplying these denominators together, giving us \( (x+2)(x-3) \). By rewriting each fraction using this shared base, we set the stage for seamless subtraction.

• Multiplying the first fraction's numerator and denominator by \(x-3\), aligns it with the common denominator.
• For the second fraction, do the same but multiply by \(x+2\) instead.

Once complete, your fractions share a uniform base, allowing for straightforward subtraction.

Simplification

Simplification in algebra involves reducing expressions to their most streamlined form. After aligning algebraic fractions under a common denominator, they often appear more complex.

This calls for algebraic simplification. It’s a sequence of steps where you combine like terms and reduce expressions to balance out the equation.

• Start by expanding all terms in each fraction's numerator, breaking them down to basic arithmetic or algebraic forms.
• In this exercise, expanding \(2x(x-3)\) results in \(2x^2 - 6x\), and \((x+1)(x+2)\) gives \(x^2 + 3x + 2\).

Subsequently, these expanded terms undergo subtraction. Pay close attention to signs to avoid simple arithmetic mistakes!

Combining like terms, such as the \(2x^2\) and \(x^2\), or the \(-6x\) and \(-3x\), refines the fraction into a simpler form, notably \(x^2 - 9x - 2\).

This minimal state is the essence of algebraic simplification.

Factoring Polynomials

Factoring polynomials is a core skill in algebra that involves breaking down expressions into products of simpler expressions. At times, the simplified result of algebraic fractions can appear factorable.

• Factoring can reveal even more reductions and often can simplify fractions further.
• It involves recognizing patterns or applying specific methods like taking out the greatest common factor (GCF), using the difference of squares, or applying special trinomials.

For example, if a numerator simplifies down to \(x^2 - 9x - 2\), explore if it factors further. Factoring helps in solving equations, simplifying further, or providing deeper insights into the structure of algebraic expressions.

However, in cases where the numerator cannot be factored meaningfully, as with \(x^2 - 9x - 2\) in this example, the expression remains as is. Knowing when and how to factor is crucial in algebra and streamlines your problem-solving process.