Simplify (D(x)(900))/(10.5+30x)

Solution:

Step 1:

Extract the common factor of $1.5$ from the denominator $10.5+30x$.

Step 1.1:

Take $1.5$ out of $10.5$. $\frac{D(x)\cdot 900}{1.5(7)+30x}$

Step 1.2:

Take $1.5$ out of $30x$. $\frac{D(x)\cdot 900}{1.5(7)+1.5(20x)}$

Step 1.3:

Pull out $1.5$ from the entire denominator. $\frac{D(x)\cdot 900}{1.5(7+20x)}$

Step 2:

Reposition $900$ to precede $D(x)$. $\frac{900\cdot D(x)}{1.5(7+20x)}$

Step 3:

Factor out $900$ from $900\cdot D(x)$. $\frac{900(D(x))}{1.5(7+20x)}$

Step 4:

Split the fraction into two parts. $\frac{900}{1.5}\cdot \frac{D(x)}{7+20x}$

Step 5:

Calculate $900$ divided by $1.5$. $600\cdot \frac{D(x)}{7+20x}$

Step 6:

Merge $600$ with $\frac{D(x)}{7+20x}$. $\frac{600D(x)}{7+20x}$

Knowledge Notes:

The problem at hand involves simplifying a mathematical expression that contains a function $D(x)$ multiplied by a constant and divided by a linear expression in $x$. The process of simplification includes factoring out common terms, rearranging terms for clarity, and performing arithmetic operations to reduce the expression to its simplest form.

The steps taken in the solution involve:

1. Factoring: This is the process of rewriting an expression as a product of its factors. In this case, $1.5$ is factored out from the terms in the denominator.

2. Rearranging terms: The multiplication of terms in the numerator is commutative, so $900\cdot D(x)$ can be rearranged as $D(x)\cdot 900$.

3. Separating fractions: The expression is split into two separate fractions to isolate the constant and the function of $x$.

4. Division: The constant terms $900$ and $1.5$ are divided to simplify the fraction.

5. Combining terms: Finally, the simplified constant is combined with the function of $x$ over the simplified denominator to present the final simplified expression.

In mathematical notation, the use of parentheses and the correct placement of terms are crucial for clarity and to avoid ambiguity. The use of LaTeX ensures that the mathematical expressions are presented in a clear and standardized form.