Problem 21 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(\frac{2 x+6}{x+3} \cdot \frac{3}{4 x}\)

Step-by-Step Solution

Answer

The simplified result is \(\frac{3}{2x}\).

Step 1: Identify the Numerators and Denominators

We are multiplying two fractions: \(\frac{2x+6}{x+3}\) and \(\frac{3}{4x}\). The numerators are \((2x+6)\) and \(3\), while the denominators are \((x+3)\) and \((4x)\).

Step 2: Multiply the Numerators

Multiply the numerators together: \((2x+6) \cdot 3\). This gives us an expression of \(6x + 18\).

Step 3: Multiply the Denominators

Multiply the denominators together: \((x+3) \cdot (4x)\). This yields \(4x(x+3)\), or expanded, \(4x^2 + 12x\).

Step 4: Simplify the Fraction

The expression is now \(\frac{6x + 18}{4x^2 + 12x}\). To simplify, factor the numerator and the denominator. The numerator \(6x + 18\) can be factored as \(6(x+3)\), and the denominator \(4x^2 + 12x\) as \(4x(x+3)\).

Step 5: Cancel Common Factors

The common factor \((x+3)\) in the numerator and the denominator can be canceled out, simplifying the fraction to \(\frac{6}{4x}\).

Step 6: Simplify Further if Possible

The fraction \(\frac{6}{4x}\) can be further simplified by dividing the numerator and the denominator by 2, resulting in \(\frac{3}{2x}\).

Key Concepts

Fraction Simplification

Simplifying fractions is like telling a shorter story. You take a fraction that might seem big and wordy, and you turn it into something smaller and easier to understand. In our exercise, we ended up with a fraction full of expressions: \( \frac{6x + 18}{4x^2 + 12x} \).

Let's get this fraction to its simplest form. The simplest form is one where the numerator and the denominator have no common factors except for 1.

We notice that both parts, top and bottom, can be shrunk by removing shared bits.

• Factor out common numbers: For the numerator \(6x + 18\), we notice 6 is common. So it becomes \(6(x+3)\).
• For the denominator \(4x^2 + 12x\), both terms are touchable by 4 and x. It factors to \(4x(x+3)\).
• Cancel out same parts: Both have \((x+3)\). We cancel it out to simplify.

After all this shrinking, we end up with \( \frac{6}{4x} \). Simplifying even more, we divide top and bottom by 2, arriving at \( \frac{3}{2x} \). That’s how we make fractions simpler and less bulky.

Factoring Expressions

Factoring is like uncovering a hidden treasure inside an expression. It involves figuring out which numbers or expressions multiply together to give us our original term.

In our mathematical adventure, we encountered expressions \(6x + 18\) and \(4x^2 + 12x\). To keep our equation neat, we needed to break these down.

Here’s how we did it:

• Find common elements: For \(6x + 18\), what number fits neatly into both 6x and 18? It's 6! So, \(6(x+3)\) is the factorized form.
• Look at \(4x^2 + 12x\). Both pieces love the company of 4 and x. Extracting these gives us \(4x(x+3)\).

This technique is essential because without it, simplifying fractions or any algebraic expressions is tricky. Factoring reveals these secrets. It's like organizing the equation closet, making sure everything is in the right place.

Algebraic Fractions

Algebraic fractions may at first look intimidating, but they're all about taking what we know about regular fractions and fitting it with algebra.

In these fractions, numerators or denominators are expressions instead of just numbers. In our exercise \( \frac{2x+6}{x+3} \cdot \frac{3}{4x} \), each fraction showcases not just numbers but a bit more: you're seeing variables and expressions.

Don’t be overwhelmed. Just like with regular numbers:

• Multiplying happens across tops and bottoms. Multiply numerators together, then denominators. This maintains our fraction reality.
• Reducing algebraic fractions involves canceling out common expressions. Once all terms are factored, shared parts can vanish just like common numbers would.

Our final outcome, \( \frac{3}{2x} \), is much friendlier to use. That’s the magic of dealing with algebraic fractions: just handle them a step at a time, always looking for common and simple outcomes!