Problem 33 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(15 x\left(\frac{x+1}{5 x}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(3x + 3\).

Step 1: Distribute the Multiplication

First, distribute the multiplication of 15x across each term inside the parentheses \( \frac{x+1}{5x} \). This means multiplying 15x by each term of the fraction.

Step 2: Simplify the Expression

Simplify the expression inside using multiplication. Multiply 15x by the numerator \(x+1\) resulting in \(15x(x+1)\), and divide by the denominator 5x: \[\frac{15x(x+1)}{5x}\]

Step 3: Cancel Common Factors

Notice that 15x can be divided by 5x, cancelling out the common factor. Divide both the numerator and the denominator by 5x to get:\[(x+1)\times 3\].

Step 4: Final Simplification

Now, simplify the expression by performing the multiplication. The result is simply:\[3(x+1) = 3x + 3\].

Key Concepts

Multiplication of expressions

In algebra, multiplication of expressions involves multiplying each part of an expression by each part of another. This is similar to distributing elements in arithmetic. In our exercise, we have a term 15x and a fractional expression \( \frac{x+1}{5x} \). Multiplication here requires us to distribute 15x to every term within the fraction.This means:

• Multiplying 15x with the numerator \( (x+1) \)
• Not forgetting to address the multiplication of denominators, even if indirectly

After multiplying, focus initially is on addressing the numerator, resulting in an expression \( 15x(x+1) \). Then, remember to keep the denominator in your final expression which is still 5x, leading to our interim result:\[ \frac{15x(x+1)}{5x} \].This intermediate step is crucial for understanding the role each part of the fraction plays and for setting up the stage for simplification.

Simplification in algebra

Simplifying algebraic expressions is about reducing an expression to its simplest form without changing its value. It often involves performing operations like distributing, combining like terms, and cancelling out when possible.In our example, the expression is initially \( \frac{15x(x+1)}{5x} \) after multiplication. Notice we have a common factor 5x in both the numerator and the denominator. To simplify:

• We divide the numerator by the 5x, which is possible by cancelling it out.
• Leaving us with the new expression: \( (x+1) \times 3 \).

At this point, focus on simplifying further by performing the remaining operations. Multiplying \((x+1)\) by 3 leads to a much simpler form \[ 3(x+1) = 3x + 3 \].Simplification in algebra helps enhance clarity and reveals the nature of the expression.

Fractional expressions

Handling fractional expressions involves both multiplication and simplification. Fractions in algebra often appear more complex due to variables in their numerators and denominators. Here's a simple method to handle them:

• Keep in mind that multiplying any term by the denominator could lead to canceling terms.
• Focus on simplifying numerators and denominators separately before cancellation.
• Always check for common factors that can be divided out to simplify.

In our exercise, we observe this process with \( \frac{15x(x+1)}{5x} \). Here, both the numerator and denominator share a 5x factor which gives us the opportunity for simplifying: - Cancel the 5x from both the numerator and denominator.This not only transforms the fraction into a much simpler component but importantly clarifies the structure of what remains, \( (x+1) \times 3 \), finalizing as \( 3x + 3 \). Understanding fractional expressions empowers you to understand large or complex expressions more easily.