Problem 80 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{x}{x-4}+\frac{5}{x+5}-\frac{11 x-8}{x^{2}+x-20}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{4x+40}{x^2+x-20}\).

Step 1: Identify Common Denominator

The common denominator for all the fractions is given by the product of individual denominators, here it is \( (x-4)(x+5) \).

Step 2: Adjusting the Fractions

Express each fraction with the common denominator, giving us: \[ \frac{x(x+5)}{(x-4)(x+5)} + \frac{5(x-4)}{(x+5)(x-4)} - \frac{11x-8}{x^2+x-20} \]

Step 3: Combine the Fractions

Now combine all the fractions by adding or subtracting the numerators: \[ \frac{x(x+5)+5(x-4)-(11x-8)}{x^2+x-20} \]

Step 4: Simplify the Expression

Simplify the numerator and denominator to get the final answer. The simplified form is \[ \frac{4x+40}{x^2+x-20} \].

Key Concepts

Common Denominator

In the world of fractions, the common denominator is the key to unlocking the simplest form of an expression. When faced with multiple fractions, finding a common denominator is a crucial step because it allows all the fractions to be combined into a single fraction, making the equation simpler to handle.

Think of the common denominator as a common language that all fractions in an equation need to speak in order to work together. In mathematical terms, it is the least common multiple of the denominators. In the exercise provided, the common denominator for the fractions \(\frac{x}{x-4}\), \(\frac{5}{x+5}\), and \(\frac{11x-8}{x^2+x-20}\) is the product of the unique factors found in their denominators, which in this case is \( (x-4)(x+5) \).

Once determined, each fraction is adjusted, or \

Combining fractions is much like making a fruit salad - you want a bit of everything in every bite. To achieve this, you need to make sure every item (or fraction) is cut (or adjusted) to the size of the common denominator. After adjusting the size, all the pieces can be put together seamlessly.

After ensuring all fractions in the exercise are speaking the same language using the common denominator \( (x-4)(x+5) \), they are able to be combined. This is done by adding or subtracting their numerators, effectively mixing them into one fraction. In mathematical form, this looks like \[ \frac{x(x+5)+5(x-4)-(11x-8)}{x^2+x-20} \]. Through this process, multiple fractions with the same denominator become one, which is easier to simplify and work with.

Algebra often involves simplifying expressions to make them more understandable and easier to work with. Simplifying an algebraic expression is akin to cleaning up your room; combining like terms and getting rid of unnecessary clutter reveals a neater, more comprehensible space (or in this case, expression).

Upon combining fractions, you are left with one fraction whose numerator often contains an algebraic expression that should be simplified. This involves expanding the products, combining like terms, and canceling out terms if possible. In the exercise, simplifying the numerator of our combined fraction \[ \frac{x(x+5)+5(x-4)-(11x-8)}{x^2+x-20} \] and factoring the common terms will reveal the truest form: \[ \frac{4x+40}{x^2+x-20} \]. This expression, now free of excess terms and noise, clearly communicates the relationship between the remaining variables and numbers.