Problem 90 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{x+3}{2 x-1}+\frac{3}{10 x^{2}-5 x} $$

Step-by-Step Solution

Answer

\(\frac{5x^2 + 15x + 3}{5x(2x-1)}\) is the simplified expression.

Step 1: Factor the Denominator

First, observe the second denominator, "10x^2 - 5x". Notice it can be factored as "5x(2x - 1)". So we rewrite the second fraction: \[ \frac{3}{10x^2 - 5x} = \frac{3}{5x(2x - 1)} \]

Step 2: Identify a Common Denominator

The common denominator must be the least common multiple of the two denominators \((2x - 1)\) and \(5x(2x - 1)\). Since \((2x - 1)\) is contained in both, the common denominator is \(5x(2x - 1)\).

Step 3: Rewrite Fractions with the Common Denominator

Rewrite each fraction with the common denominator:For the first fraction, multiply both the numerator and denominator by \(5x\):\[ \frac{x+3}{2x-1} \times \frac{5x}{5x} = \frac{5x(x+3)}{5x(2x-1)} \]The second fraction is already in the required form:\[ \frac{3}{5x(2x - 1)} \]

Step 4: Combine the Fractions

Now you can combine the fractions:\[ \frac{5x(x+3) + 3}{5x(2x-1)} \]

Step 5: Simplify the Numerator

Simplify the expression in the numerator:Distribute \(5x\) in the first term:\[ 5x(x + 3) = 5x^2 + 15x \]Combine it with the second term:\[ 5x^2 + 15x + 3 \]

Step 6: Check for Further Simplification

Since there are no common factors in the numerator and denominator, the expression cannot be simplified further. The simplified expression is:\[ \frac{5x^2 + 15x + 3}{5x(2x-1)} \]

Key Concepts

Simplifying Fractions

Simplifying algebraic fractions is all about making them as easy to handle as possible. Think of it like reducing a recipe to fewer ingredients while still keeping the essence the same. This involves finding if the numerator (top part) and the denominator (bottom part) share any common factors that can be "cancelled" out to simplify the expression.

In the context of simplification, always look for:

• Common terms in both the numerator and the denominator.
• The possibility of factoring, which we'll discuss more later, to make the expression simpler.

For example, if you had \( \frac{10x^2 + 20x}{5x}\), you would notice each term in the numerator shares \( 5x \) as a factor with the denominator. You start by dividing each term by the common factor \( 5x \), reducing the fraction to:\(2x + 4\).

Simplifying fractions often sets the base for more complex operations such as addition, subtraction, and solving equations by making calculations easier and more straightforward.

Factoring

Factoring is like breaking down numbers or expressions into multiples that come together to form the original number or expression. This is an essential skill in algebra because it allows you to simplify and solve equations with ease.

When dealing with algebraic fractions, you should always inspect (especially the denominators) for opportunities to factor them out.Consider the expression:\(10x^2 - 5x\).

A good first step is to find the greatest common factor (GCF), which, in this case, is \(5x\). Factoring this out gives you:\[10x^2 - 5x = 5x(2x - 1)\].

By factoring, arithmetic becomes simpler and more manageable. It plays a fundamental role in finding common denominators and simplifying expressions. Remember to always do factoring when possible, as it lays the groundwork for solving more complicated algebra problems.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions. Just like with regular, numerical fractions, the aim is tofind a shared base so you can directly compare and combine the fractions.

In algebra, the goal is slightly different. You're looking for a common denominator based on the expressions given in the denominators, typically by finding the least common multiple (LCM). For example:

• Take the denominators \((2x - 1)\) and \(5x(2x - 1)\).
• The expression \((2x - 1)\) is contained in both, so the LCM, or the common denominator, is \(5x(2x - 1)\).

By rewriting each fraction with this common denominator, they can be combined:

For the first fraction:\[\frac{x+3}{2x-1} \times \frac{5x}{5x} = \frac{5x(x+3)}{5x(2x-1)}\]

Now, since the second fraction was already using the common denominator, you simply replace and combine:\[\frac{5x(x+3) + 3}{5x(2x-1)}\].

By understanding and applying the concept of a common denominator, you can effectively work with and solve algebraic fractions.