Problem 44 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{4}{x}-\frac{3}{x+3} $$

Step-by-Step Solution

Answer

The solution to the expression after addition or subtraction is \(\frac{x + 12}{x^2 + 3x}\)

Step 1: Identify the Least Common Denominator

Inspecting the two fractions \(\frac{4}{x}\) and \(\frac{3}{x + 3}\), the denominators do not match. To add or subtract these fractions, they will need to have a common denominator. In this case, the least common denominator would be \(x(x + 3)\).

Step 2: Adjust fractions to have common denominator

Adjust the original fractions to have the common denominator. Multiply the first fraction by \(\frac{x + 3}{x + 3}\) and the second fraction by \(\frac{x}{x}\). This results in, \(\frac{4(x + 3)}{x(x + 3)} -\frac{3x}{x(x + 3)}\).

Step 3: Simplify the Numerators

Perform the operations in the numerators to simplify the fractions. This will result in \(\frac{4x + 12}{x^2 + 3x} - \frac{3x}{x^2 + 3x}\). Since both fractions have the same denominator, they can be merged: \(\frac{4x + 12 - 3x}{x^2 + 3x}\).

Step 4: Further Simplify the Fraction

Further simplify the numerator by subtracting the like terms which results in \(\frac{x + 12}{x^2 + 3x}\).

Key Concepts

Least Common Denominator

When dealing with fractions, especially rational expressions that have different denominators, finding a common ground for them to interact through addition or subtraction is essential. This 'common ground' is known as the Least Common Denominator (LCD). The LCD is the smallest expression that is a multiple of both denominators. It ensures that when you adjust your fractions, you don’t change their value, merely their form.

For fractions like \(\frac{4}{x}\) and \(\frac{3}{x+3}\), the denominators \(x\) and \(x+3\) are not the same, making direct addition or subtraction impossible. Hence, we find the LCD by multiplying the distinct denominators together, resulting in \(x(x+3)\). This product embodies the smallest expression containing all the factors needed by both original denominators. Once the LCD is identified, fractions can be adjusted to have this common denominator, paving the way for effective addition or subtraction.

Adding and Subtracting Fractions

Once fractions share a common denominator, adding or subtracting them becomes straightforward. Let's consider our adjusted fractions: \(\frac{4(x+3)}{x(x+3)}\) and \(\frac{3x}{x(x+3)}\). With both fractions now featuring the LCD as their base, you can easily combine them.

The general rule involves working solely with the numerators. For addition, simply sum up the numerators over the shared denominator. In subtraction, as with our expression \(\frac{4(x+3)}{x(x+3)} - \frac{3x}{x(x+3)}\), subtract the second numerator from the first. This operation brings us to a singular fraction that looks neat: \(\frac{4x + 12 - 3x}{x^2 + 3x}\). Enumerating the fractions requires attention to sign and simplification as we move forward.

Simplifying Fractions

After performing addition or subtraction on fractions, the next goal is simplification. Simplifying fractions makes them easier to interpret and use in further calculations. In the expression \(\frac{4x + 12 - 3x}{x^2 + 3x}\), we focus first on the numerator.

Start by combining like terms, where applicable. For our example, \(4x\) and \(-3x\) are like terms. Together, they simplify to \(x\), thus transforming the numerator to \(x + 12\). The denominator remains as \(x^2 + 3x\) since further simplification needs a careful examination for factor commonality or reducible terms.

Sometimes, simplified expressions can still be reduced further depending on factorization or canceling out common factors between the numerator and the denominator. However, in our instance, the fraction has reached its simplest form. This process not only streamlines the expression but also provides a clearer view of its components, aiding in easier and more accurate application in mathematical scenarios.