Problem 70 - Algebra and Trigonometry

Question

Simplify the compound fractional expression. $$ \frac{\frac{x-3}{x-4}-\frac{x+2}{x+1}}{x+3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5}{(x-4)(x+1)(x+3)} \).

Step 1: Identify the Compound Fractions

The given expression is a compound fraction with a main numerator \( \frac{x-3}{x-4} - \frac{x+2}{x+1} \) and denominator \( x+3 \). Our goal is to simplify this compound fraction.

Step 2: Simplify the Numerator

Simplify the numerator \( \frac{x-3}{x-4} - \frac{x+2}{x+1} \) by finding a common denominator. The common denominator of \( x-4 \) and \( x+1 \) is \( (x-4)(x+1) \). Rewriting each fraction to have this common denominator, we get:\[ \frac{(x-3)(x+1) - (x+2)(x-4)}{(x-4)(x+1)} \].

Step 3: Expand and Simplify the Numerator Expression

Expand \((x-3)(x+1)\) and \((x+2)(x-4)\):- \((x-3)(x+1) = x^2 - 3x + x - 3 = x^2 - 2x - 3\)- \((x+2)(x-4) = x^2 + 2x - 4x - 8 = x^2 - 2x - 8\)Now subtract: \[ (x^2 - 2x - 3) - (x^2 - 2x - 8) = x^2 - 2x - 3 - x^2 + 2x + 8 = 5 \].So, the numerator simplifies to \( \frac{5}{(x-4)(x+1)} \).

Step 4: Simplify the Overall Expression

Now we have a simplified numerator \( \frac{5}{(x-4)(x+1)} \) and the denominator \( x+3 \). This overall can be written as: \[ \frac{5}{(x-4)(x+1) \cdot (x+3)} \].

Step 5: Final Simplified Expression

The expression is fully simplified with no further factorable components or simplifications applicable to \( \frac{5}{(x-4)(x+1)(x+3)} \). Thus, the final simplified expression is \( \frac{5}{(x-4)(x+1)(x+3)} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is all about reducing the fraction to its simplest form. Imagine you have a big piece of pizza and you want to get it into a smaller, more manageable size. That’s what simplifying does to a fraction.
In mathematical terms, simplifying involves dividing both the numerator (the top part) and the denominator (the bottom part) of the fraction by their greatest common divisor (GCD). This process leaves you with the simplest equivalent fraction, where the numerator and denominator have no common factors other than 1.

• For example, think of the fraction \( \frac{8}{12} \). The GCD of 8 and 12 is 4. Divide both by 4 to get \( \frac{2}{3} \), which is the simplified form.

In the case of compound fractions, such as those we are dealing with in this exercise, simplifying can often mean ensuring that the fractions inside the fraction are reduced or combined neatly using common denominators or by canceling out terms.

Algebraic Expressions

Algebraic expressions are a way of representing numbers and operations in a generalized form using variables, constants, and operations. Variables are symbols, often letters, such as \(x\) or \(y\), that stand in for unknown numbers.
These expressions can be very simple, like \(3x\) or as complex as the fractions involving \( (x-3)\) and \((x+2)\) seen in our exercise. Essentially, algebraic expressions can include addition, subtraction, multiplication, division, and exponentiation between variables and numbers.

• Example: In the expression \( 2x + 3y - 5 \), \( 2x \) and \( 3y \) are terms where \( 2 \) and \( 3 \) are coefficients, and \(x\) and \(y\) are variables.

By working with algebraic expressions, we perform operations such as expanding, factoring, and simplifying, which help us solve equations or simplify complex algebraic fractions like the one reviewed.

Common Denominator

Finding a common denominator is an essential step in simplifying complex fractions because it allows us to combine fractions that have different denominators.
When two fractions are subtracted or added together, a common denominator helps make the denominators the same so their numerators can be added or subtracted directly. In simpler terms, think of it as having two pieces of pie that need to be on plates of the same size before you can decide how much you actually have together.
To find a common denominator:

• Identify the denominators of the fractions involved. For example, in \( \frac{x-3}{x-4} \) and \( \frac{x+2}{x+1} \), the denominators are \(x-4\) and \(x+1\).
• Multiply the denominators to construct a common denominator: \((x-4)(x+1)\).
• Rewrite each fraction so that they share this new common denominator.

This not only simplifies the process but is a powerful technique often used in algebraic equations and rational expressions, aiding greatly in the step-by-step simplification of these intertwined expressions.