Problem 9 - Basic Technical Mathematics with Calculus

Question

In Exercises \(5-12,\) multiply the numerator and the denominator of each fraction by the given factor and obtain an equivalent fraction. $$\frac{2}{x+3}(\text { by } x-2)$$

Step-by-Step Solution

Answer

The equivalent fraction is \(\frac{2x-4}{(x+3)(x-2)}\).

Step 1: Understand the Given Fraction and Factor

The problem provides the fraction \(\frac{2}{x+3}\) and asks us to multiply both the numerator and the denominator by a given factor \(x-2\). This means we will form a new equivalent fraction by applying this multiplication.

Step 2: Multiply the Numerator by the Factor

Multiply the numerator 2 by the factor \(x-2\). This gives us \(2(x-2) = 2x - 4\).

Step 3: Multiply the Denominator by the Factor

Multiply the denominator \(x+3\) by the factor \(x-2\). This gives us \((x+3)(x-2)\).

Step 4: Write the Equivalent Fraction

The equivalent fraction is formed by placing the new numerator and new denominator together. This yields: \[\frac{2x-4}{(x+3)(x-2)}\]. We do not need to further simplify unless it’s requested or readily possible.

Key Concepts

Numerator and Denominator

When dealing with fractions, understanding the roles of the numerator and denominator is crucial. A fraction consists of two parts:

• Numerator: The top part of the fraction. It represents how many parts of the whole are being considered.
• Denominator: The bottom part of the fraction. It indicates how many total parts make up the whole.

For example, in the fraction \(\frac{2}{x+3}\), "2" is the numerator, which tells us we are considering 2 parts, and "\(x+3\)" is the denominator, which indicates the whole has \(x+3\) parts.

The relationship between numerator and denominator affects how the fraction behaves. If both parts are multiplied by the same factor, the size of the pieces changes, but the overall value of the fraction remains unchanged. This is the heart of creating equivalent fractions.

Equivalent Fractions

Equivalent fractions are fractions that may look different but represent the same value or proportion of a whole.

If you multiply or divide both the numerator and the denominator of a fraction by the same non-zero number, the resulting fraction is equivalent to the original. This is because you are scaling both parts by the same factor, maintaining the ratio between them.
For instance, let's consider the fraction \(\frac{2}{x+3}\). When we multiply both the numerator and denominator by \(x-2\), we get the fraction \(\frac{2x-4}{(x+3)(x-2)}\). Despite the fact that these two fractions look different, they are equivalent as they represent the same portion of the whole.

• It's essential to note that equivalent fractions maintain balance between numbers and variables.
• Creating equivalent fractions is a useful technique for simplifying expressions or solving equations.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators together, then multiplying the denominators together, and finally, placing the results into a new fraction.

When faced with a problem like multiplying the fraction \(\frac{2}{x+3}\) by a factor such as \(x-2\), we apply this process by:

• Multiplying the numerator \(2\) by \(x-2\), which results in \(2x - 4\).
• Multiplying the denominator \(x+3\) by \(x-2\), giving us \((x+3)(x-2)\).

This operation results in the formation of a new fraction, \(\frac{2x-4}{(x+3)(x-2)}\), which is known as an equivalent fraction since it reflects the same ratio or proportion as the original.
Understanding how to multiply fractions is handy in algebra, especially when working with expressions containing variables, helping to streamline and simplify complex problems.